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Gaps between non-consecutive primes
In the last few days I dug my fangs into gaps between primes p[SUB]n[/SUB] and p[SUB]n+k[/SUB] (with k=1 these are the usual well-known prime gaps, for k=2 see A[OEIS]144103[/OEIS], for k=3 see A[OEIS]339943[/OEIS], for k=4 see A[OEIS]339944[/OEIS]).
This can be seen as part of the effort to further improve the amount of empirical data related to prime gaps. Recently I found the paper [URL]https://arxiv.org/abs/2011.14210[/URL] (Abhimanyu Kumar, Anuraag Saxena: Insulated primes), which makes some predictions regarding k=2, but is based on quite limited empirical study. Here's a tidbit of data of especially large gaps for k=1..19 and p<6*10[SUP]12[/SUP]: [CODE] k CSG_max * p_n p_n+k 1 0.7975364 2614941710599 2614941711251 2 0.8304000 5061226833427 5061226834187 3 0.8585345 5396566668539 5396566669381 4 0.8729716 4974522893 4974523453 (largest CSG_max thus far) 5 0.8486459 137753857961 137753858707 6 0.8358987 5550170010173 5550170011159 7 0.8396098 3766107590057 3766107591083 8 0.8663070 11878096933 11878097723 9 0.8521843 1745499026867 1745499027983 10 0.8589305 5995661470529 5995661471797 11 0.8467931 5995661470481 5995661471797 12 0.8347906 5995661470529 5995661471893 13 0.8439277 5995661470529 5995661471977 14 0.8312816 5995661470481 5995661471977 15 0.7987377 5995661470471 5995661471977 16 0.7901341 5568288566663 5568288568217 17 0.7632862 396016668869 396016670261 18 0.7476038 396016668833 396016670261 19 0.7560424 968269822189 968269823761[/CODE]* A version of the Cramér-Shanks-Granville ratio. Only a quick spreadsheet formula, this could probably use some fine tuning[SUP]1)[/SUP], but for the time being, in this table [$]CSG = \Large \frac{gap}{(\log \frac{p_n+p_{n+k}}{2} +k-1)^2}[/$] [SUP]1)[/SUP] I'd prefer something like M (the "merit") = Gram(p[SUB]n+k[/SUB])-Gram(p[SUB]n[/SUB])-k+1 where Gram(x) is Gram's version of Riemann's pi(x) approximation, and CSG = M[SUP]2[/SUP]/gap - pending negotiations... Calculations will have reached p ~ 7*10[SUP]12[/SUP] by tomorrow, and additionally for k=2 with p ~ 16*10[SUP]12[/SUP]. Not terribly fast, I admit. Does anybody know of any further work on this topic? |
For each k, what are the first few gaps with record CSG ratio? This is very interesting.
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Greetings Bobby,
I'd like to run these numbers through Pari again before posting more inconsistent/approximate numbers. The formula with the term "-k+1" (see [SUP]1)[/SUP] from previous post) is only working properly when CSG = max(0,M)[SUP]2[/SUP]/gap since M can be negative (because of the aforementioned term). Working out details like these takes me inordinately long... Good news is, for p = 8,281,634,108,677 and k = 19, I get a CSG > 1 with the rough-and-ready version of the fine-tuned formula: gap = 1812, M = gap/log(p+gap/2)-18 ~ 42.918 (there are 60.918 primes on average in a range of 1812 integers, i.e. 42.918 more than the 18 that are actually between the bounding primes), and CSG = M[SUP]2[/SUP]/gap ~ 1.0165. With p that large, there won't be much of a difference anymore when using Gram(x) in the calculation of CSG. |
When n=3, a big gap seems like 35617, 35671, 35677, 35729. There is a gap of 54 between 35617 and 35671, which is big for numbers of that size. After the gap of 6 between 35671 and 35677, there is another big gap of 52 between 35677 and 35729. Therefore, the 3-gap between 35617 and 35729 is a surprisingly large prime gap.
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[QUOTE=Bobby Jacobs;593106]When n=3, a big gap seems like 35617, 35671, 35677, 35729.[/QUOTE]
It doesn't only seem like a big gap, it's listed in A339943 as a(56), since 56=(35729-35617)/2. I have a lot of data ready for submission, it just takes me longer to actually submit it, my schedule is pretty clogged at the moment... |
I looked at A115401, the record gaps between primes 3 apart, and it turns out that the gap of 112 between 35617 and 35729 is very big. The sequence starts out smoothly. After the initial 5, every even number from 8 to 36 is in the sequence. There are not many even numbers missing up to 68. Then, it jumps to 78, 84, and a really big leap to 112. That corresponds to the 35617, 35729 gap. It is an enormous prime gap!
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I have uploaded some data for posterity, differences d between primes p[SUB]n[/SUB] and p[SUB]n+k[/SUB] for k <= 130 and d <= 740, see 2nd link for A[OEIS]086153[/OEIS].
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Some data in the attachment, just to show off.
The interested reader might also like to check, for instance, the differences p(n+42)-p(n) for p in the range [327076775000..327076783000]. Makes for a nice graph. And this related result, 100 primes in the range p+[1..8349] while there are no primes in q+[1..8349], with q < p, still appears to be unmatched: [URL]https://www.mersenneforum.org/showpost.php?p=479832&postcount=86[/URL] Excuse my being a bit cocky today :wink: |
[QUOTE=mart_r;595755]Some data in the attachment, just to show off.
The interested reader might also like to check, for instance, the differences p(n+42)-p(n) for p in the range [327076775000..327076783000]. Makes for a nice graph. And this related result, 100 primes in the range p+[1..8349] while there are no primes in q+[1..8349], with q < p, still appears to be unmatched: [URL]https://www.mersenneforum.org/showpost.php?p=479832&postcount=86[/URL] Excuse my being a bit cocky today :wink:[/QUOTE] Cocky! |
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[QUOTE=robert44444uk;595925]Cocky![/QUOTE]
What? What I wrote did look a little conceited to me:smile: [QUOTE=mart_r;595755]The interested reader might also like to check, for instance, the differences p(n+42)-p(n) for p in the range [327076775000..327076783000]. Makes for a nice graph. [/QUOTE] That is, using a certain style of graph and a little imagination | | V |
[QUOTE=mart_r;595952]What? What I wrote did look a little conceited to me:smile:
[/QUOTE] Nah, not really, Excellent work as always mart_r |
New Year's Eve consolidation
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Some data for maximal gaps in the file in close proximity. [SIZE=1][COLOR=LemonChiffon]In the next update, I'll include the data for p=2, I promise.[/COLOR][/SIZE]
There are three primes (well, actually, 54 primes:) that await discovery, for k=16 / d=76 k=17 / d=82 k=18 / d=84 And possibly feasible for J. Wroblewski and R. Chermoni: k=19 / d=86 and d=88 k=20 / d=90 and d=92 As a by-product, a puzzle: Given x, find the next three consecutive primes >= x. Denote the two gaps between them g[SUB]1[/SUB] and g[SUB]2[/SUB], and let g[SUB]1[/SUB] >= g[SUB]2[/SUB]. Let r = g[SUB]1[/SUB]/g[SUB]2[/SUB]. As x becomes larger, the geometric mean r[SUB]gm[/SUB] of values of r also become larger. Find an asymptotic function f(x) ~ r[SUB]gm[/SUB]. |
Staking claims
A measurably unusual scarcity of primes appears between 6,215,409,275,042 and 6,215,409,279,556 - there are only 83 primes in-between, just a little over half as many as expected on average, and the associate CSG value is 1.0944363.
The year starts off pretty well. |
[QUOTE=mart_r;595952]
[QUOTE=robert44444uk;595925]Cocky![/QUOTE] What? What I wrote did look a little conceited to me:smile: [/QUOTE] This conversation does not make sense to me. It first seems like Robert is agreeing with you that you are being cocky. However, your response acts like he is not agreeing. Then, in the next post, he says that you are not cocky. What is going on? |
[QUOTE=Bobby Jacobs;597473]What is going on?[/QUOTE]
Building mountains out of molehills, I guess :smile: I thought Robert was making fun of me ... one word responses can be confusing, maybe it was a misunderstanding on my part. Those language barriers... Any suggestions on whether I should rather continue to search larger primes for k<=109, or to look at larger values of k? |
Hi all
I did a bit of searching around the average merit of 100 gaps in the range of 101 primes, and my best performance is (I've checked up to 9.675e11): Gap=4354 Average merit=1.622541804, from prime 450867605017 to 450867609371 My method takes the average gap to be [g/ln(p1)+g/ln(p101)]/(2*100) where g is, in this case 4354 At the other end of the spectrum, the following range: Gap=1554 Average Merit=0.584366417 from prime 354120798439 to 354120799993 |
[QUOTE=mart_r;592465]
Here's a tidbit of data of especially large gaps for k=1..19 and p<6*10[SUP]12[/SUP]: [CODE] k CSG_max * p_n p_n+k 1 0.7975364 2614941710599 2614941711251 2 0.8304000 5061226833427 5061226834187 3 0.8585345 5396566668539 5396566669381 4 0.8729716 4974522893 4974523453 (largest CSG_max thus far) 5 0.8486459 137753857961 137753858707 6 0.8358987 5550170010173 5550170011159 7 0.8396098 3766107590057 3766107591083 8 0.8663070 11878096933 11878097723 9 0.8521843 1745499026867 1745499027983 10 0.8589305 5995661470529 5995661471797 11 0.8467931 5995661470481 5995661471797 12 0.8347906 5995661470529 5995661471893 13 0.8439277 5995661470529 5995661471977 14 0.8312816 5995661470481 5995661471977 15 0.7987377 5995661470471 5995661471977 16 0.7901341 5568288566663 5568288568217 17 0.7632862 396016668869 396016670261 18 0.7476038 396016668833 396016670261 19 0.7560424 968269822189 968269823761[/CODE]..... Does anybody know of any further work on this topic?[/QUOTE] I confirm Marts values for 17,18,19 as the largest average merits between 18,19 and 20 primes respectively, It is worth looking at the minimum value found to date for these, as no-one has found the relevant all prime k-tuple at these sizes. Where 2 are listed, it shows the smallest gap and the smallest average merit in the gap. [CODE] n gap p(n) p(n+k) ave merit 17 98 341078531681 341078531779 0.21708 18 114 1054694671669 1054694671669 0.22877 18 110 43440699011 43440699121 0.24948 19 126 1085806111031 1085806111157 0.23929 19 120 31311431897 31311432017 0.26134 [/CODE] |
Here are some results for 20..25
Small average merits and gaps: [CODE] n gap p(n) p(n+k) ave merit checked to 20 138 2037404713403 2037404713541 0.243448948 2.80E+12 20 136 1085806111021 1085806111157 0.245369164 21 144 2037404713397 2037404713541 0.241936843 2.81E+12 22 160 2037404713381 2037404713541 0.256599682 2.81E+12 22 156 325117822691 325117822847 0.267506235 23 174 2766595321597 2766595321771 0.264069002 2.81E+12 24 180 220654442209 220654442389 0.287137792 1.24E+12 25 190 220654442209 220654442399 0.290966296 8.22E+11 [/CODE] And large, tested up to the same values, so it looks like the 24 and 25 records may go - no doubt somewhere in mart_r's file: [CODE] n gap p(n) p(n+k) ave merit 20 1582 968269822189 968269823771 2.866069068 21 1630 968269822189 968269823819 2.812408994 22 1680 968269822189 968269823869 2.766921063 23 1756 2137515911737 2137515913493 2.689187618 24 1740 752315299717 752315301457 2.651169565 25 1780 628177622389 628177624169 2.62091465 [/CODE] |
[QUOTE=mart_r;597502]Building mountains out of molehills, I guess :smile:
I thought Robert was making fun of me ... one word responses can be confusing, maybe it was a misunderstanding on my part. Those language barriers... Any suggestions on whether I should rather continue to search larger primes for k<=109, or to look at larger values of k?[/QUOTE] I already did a bit of work at k=1000 but I might concentrate at k=200 and 500 and see where that goes |
[QUOTE=mart_r;595755]
And this related result, 100 primes in the range p+[1..8349] while there are no primes in q+[1..8349], with q < p, still appears to be unmatched: [URL]https://www.mersenneforum.org/showpost.php?p=479832&postcount=86[/URL] [/QUOTE] This is much harder than I anticipated - it really is an outstanding result. I have started to look at the next obvious candidate starting from 3483347771*409#/30 - 7016 (merit >39). I have only achieved 67 primes so far (after about 30 minutes of checking), so I am wondering if this can ever get to 100 primes |
Thanks for your support!
Your results for maximum average merits are in accordance with my results in post # 12. I didn't look for minimum average merits as they are theoretically covered by the minimum widths of k-tuplets. But some clusters are missing, see also post # 12. However, more data is always welcome! [QUOTE=robert44444uk;597586]This is much harder than I anticipated - it really is an outstanding result. I have started to look at the next obvious candidate starting from 3483347771*409#/30 - 7016 (merit >39). I have only achieved 67 primes so far (after about 30 minutes of checking), so I am wondering if this can ever get to 100 primes[/QUOTE] Though the difference seems little (merit 39.62 vs. 41.94), it's several times as hard to fill the gap with 100 primes larger than those surrounding the gap. I'd have to check the stats, but an admissible 1886-tuplet pattern (minimum width 15899) with no factors < 400-ish would be a good start for the search. |
[QUOTE=mart_r;597600]
Though the difference seems little (merit 39.62 vs. 41.94), it's several times as hard to fill the gap with 100 primes larger than those surrounding the gap. I'd have to check the stats, but an admissible 1886-tuplet pattern (minimum width 15899) with no factors < 400-ish would be a good start for the search.[/QUOTE] I'm trying to understand the approach. I've found a 1886-tuplet pattern width 15898 from the internet,[URL="https://math.mit.edu/~primegaps/tuples/admissible_1886_15898.txt"]https://math.mit.edu/~primegaps/tuples/admissible_1886_15898.txt[/URL] so is the idea to get a Chinese Remainder (C) based on mods of primes <400, referenced the start prime of the large gap (P), and then to prp from P+n*C to P+n*C+15900, n integer? Or is there further sieving to do? Are the Chinese mods gotten by a greedy algorithm? Is such a large Chinese potentially inferior to a much smaller Chinese (c) based around say 1000-tuplet where, if the prime count was high after testing, then it could be tested over the whole range. I'm thinking this trades off the greater chance of primes with ranges close to P, i.e. at P+c*n against the low chance at P+C*n |
[QUOTE=robert44444uk;597644]I'm trying to understand the approach.
I've found a 1886-tuplet pattern width 15898 from the internet,[URL="https://math.mit.edu/~primegaps/tuples/admissible_1886_15898.txt"]https://math.mit.edu/~primegaps/tuples/admissible_1886_15898.txt[/URL] so is the idea to get a Chinese Remainder (C) based on mods of primes <400, referenced the start prime of the large gap (P), and then to prp from P+n*C to P+n*C+15900, n integer? Or is there further sieving to do? Are the Chinese mods gotten by a greedy algorithm? Is such a large Chinese potentially inferior to a much smaller Chinese (c) based around say 1000-tuplet where, if the prime count was high after testing, then it could be tested over the whole range. I'm thinking this trades off the greater chance of primes with ranges close to P, i.e. at P+c*n against the low chance at P+C*n[/QUOTE] so if I have done this right, the CRT offset is 49268213492433141497814341275312197605721177068674522156228345708919204704299688530737031645921153516711274856551464412837807539813310218115111791871010488468153 This is based on the mods of primes to 400 that never produce 0mod(the prime) for all values in the admissible sets. So 1mod2, 1mod3, 3mod5, 4mod7... this is approx. 5e160, compared to the prime at the start of the gap of 15900, which is 2e174, so it looks fine to play around with. |
I get the same CRT offset with the pattern you linked to, so that's correct.
For my result, I didn't bother too much about sieving and just tested n*p#+c+x for primes, incrementing n when not enough primes were found above a customized threshold for x; something should be gained by applying an appropriate sieving technique. Up to 479#, there's only one open residue class for each prime, so it should merely be checked that not too many potential coprimes are cancelled out by the sieve. Just for fun, here are offsets for some larger p#: [CODE]n*401#+ 39513451711353368972101707142676951932015103896038867201264946985487496815918734804213619530781605639321227211666671723990836697616026451458080515468952387537444673 n*409#+ 5300963833569209940057949054368721853533208296343484993741509551411018530966721606193800807153951251943913507869529328702625642328421257335999756718592000205492358083 n*419#+ 906110212932116653575732557665488316402090982314903912519073620662591471401157669028755216511381275347162712814920340507230752131956051717149856040611426373130703347023 [/CODE] |
I'm doing something wrong I think, although I am not sure (maybe mart_r could check)
Average number of primes in a range of x=15900 integers from a = 3483347771*409#/30 is = 15900/ln(a) or approx 39.65, given an average gap of 401. Average found number primes for n from 0 to 100 in n*p#+c+x, is 40.01 with a max prime count of 54 at n=93. c offset: 49268.... I am surprised to see such a small average pickup it is well within the bounds of statistics to be zero effect. if I am doing this right I am not sure the method pays off. |
[QUOTE=robert44444uk;597739]I'm doing something wrong I think, although I am not sure (maybe mart_r could check)
Average number of primes in a range of x=15900 integers from a = 3483347771*409#/30 is = 15900/ln(a) or approx 39.65, given an average gap of 401. Average found number primes for n from 0 to 100 in n*p#+c+x, is 40.01 with a max prime count of 54 at n=93. c offset: 49268.... I am surprised to see such a small average pickup it is well within the bounds of statistics to be zero effect. if I am doing this right I am not sure the method pays off.[/QUOTE] I was doing this very wrong, silly me. I think I was wrong to start at the deficient primorial 409#/30, I should have started with the primorial 409#, with a lower multiplier, in this case 3483347771/30 = 116111593 rounded up. The I don't multiply the offset c, I add one each time to n. My results for the first 100 n above 116111593 shows an average of 48.62 with a maximum of 63. That's more like it! |
[QUOTE=robert44444uk;597555]I already did a bit of work at k=1000 but I might concentrate at k=200 and 500 and see where that goes[/QUOTE]
500 results - tested to approx. 1.1e12: Largest gap: 16690 Average merit: 1.229114171 First prime: 622973626447 Smallest gap: 10306 Average merit: 0.795722241 First prime: 177726413581 200 results: tested to approx. 1.39e12 Largest: 7338 Average merit: 1.414200628 First prime:185067242119 Smallest: 3646 Average merit: 0.694714106 First prime: 249072607711 100 results: tested to approx. 2.4e12 Largest: 4540 Average merit: 1.642701445 First prime: 1006401165853 Smallest: 1640 Average merit: 0.580014238 First prime: 1904361666929 |
[QUOTE=robert44444uk;597747]I was doing this very wrong, silly me.
I think I was wrong to start at the deficient primorial 409#/30, I should have started with the primorial 409#, with a lower multiplier, in this case 3483347771/30 = 116111593 rounded up. The I don't multiply the offset c, I add one each time to n. My results for the first 100 n above 116111593 shows an average of 48.62 with a maximum of 63. That's more like it![/QUOTE] After 20 hours of processing I'm afraid that I can beat 87 primes in a range of 15900 - I'll continue the search though In n*p#+c+x, where p = 409, c = 492682..., x from 0 to 15900, then 87 primes are at n = 117575956 118482688 |
[QUOTE=robert44444uk;597768]
100 results: tested to approx. 2.4e12 Largest: 4540 Average merit: 1.642701445 First prime: 1006401165853 Smallest: 1640 Average merit: 0.580014238 First prime: 1904361666929[/QUOTE] As an aid to the factorials and offsets approach, it is relatively simple to show that there is no range of 100 primes p1..p100 at relatively small p1 where p1 is larger than any gap smaller than p1. In the exhaustive check of gaps between 101 primes ( a proxy for 100) highlighted with p1 <2.4e12, no range has an average merit of < 0.58, which equates to requiring a gap, where p1 is less than 2.4e12 whose merit is > 58. As the largest merit ever found is not even 42, there is the basis for the proof. |
[QUOTE=robert44444uk;597925]After 20 hours of processing I'm afraid that I can beat 87 primes in a range of 15900 - I'll continue the search though
In n*p#+c+x, where p = 409, c = 492682..., x from 0 to 15900, then 87 primes are at n = 117575956 118482688[/QUOTE] It took me a couple of days, so I do think you have a chance to beat me at my own game :smile: I was also trying to squeeze even more primes into an interval of 8348, about two years ago, but a couple more days of searching and I didn't get past about 94 or 95 primes. |
A slight improvement in the prime count for n in n*p#+c+x, where p = 409, c = 492682..., x from 0 to 15900
123733011 91 120673847 88 121848887 88 In terms of sieves, I found it was useful to break down the range x into four even parts and set cumulative targets for each range. I found at least 8 values at 50 or more primes at half way with a top value of 53. I do feel that it would be better to concentrate more on the 41 merit gap, maybe using a few new tweaks to get to the 101 level. Almost 42 merit is totally different to 39 in terms of space. |
A couple more, getting closer, but not that close
124977806 92 125316443 90 |
[QUOTE=robert44444uk;598157]A couple more, getting closer, but not that close
124977806 92 125316443 90[/QUOTE] Any chance you can still tweak the algorithm a bit in your favor? Some possibly useful ideas, theory-wise: Let [$]p_n[/$] be large (well, say, > 10[SUP]6[/SUP]), let [$]p_{n+k}[/$] be the k-th prime after [$]p_n[/$] (k < [$]\sqrt{p_n}[/$], just to be safe). [$]g = p_{n+k} - p_n[/$] [$]m = G(p_{n+k})-G(p_n)-k+1[/$], G(x) being the formula for the blue line in the graph in [URL]http://www.primefan.ru/stuff/primes/table.html#theory[/URL] [$]CSG^* = \frac{m \cdot |m|}{g}[/$] [$]m^* = CSG^* \cdot \log p_{n+k}[/$] One may be inclined to expect the distribution of [$]m^*[/$] as behaving like the ones for the merits of usual prime gaps. This may even be half-way right, as experimental data suggests, for example these 1,751,000 samples of intervals of length 10[SUP]5[/SUP] with p in the vicinity of 34*10[SUP]12[/SUP] can be turned into this (similar results for other parameters): [CODE]m*< #times r (#/total) log(1-r) 1 1581721 0.903324386 -2.336394091 1) 2 1695327 0.968205026 -3.448447043 3 1731016 0.988587093 -4.473010379 4 1743358 0.995635637 -5.434282983 5 1747999 0.998286122 -6.368996766 6 1749835 0.999334666 -7.315221245 7 1750528 0.999730440 -8.218718626 8 1750830 0.999902913 -9.239899174 9 1750932 0.999961165 -10.15618991 10 1750978 0.999987436 -11.28465516 11 1750990 0.999994289 -12.07311252 12 1750995 0.999997144 -12.76625970 13 1750997 0.999998287 -13.27708532 14 1750998 0.999998858 -13.68255043 15 1750999 0.999999429 -14.37569761 16 1751000 1 [/CODE]1) Note: for usual gaps and merit < 1, log(1-r) would statistically be around -1, but [$]m^*[/$] is < 0 around half of the time, and due to the special treatment the distribution in the low range of [$]m^*[/$] is somewhat... different. Darn, I need to find the time to catch up on Gaussian/Poisson/... distribution measures. Would that lead to a way to conjecture that the gaps between non-consecutive primes are bounded by a constant times [$](\log(p)+k) \cdot \log(p)[/$] ? Or am I thinking way too complicated? |
The closest I have gotten to 100 primes following 1571162669*193#+129568114146274965711541776666046371290799466131684641935400586161726498035577 is 95 primes, within a period of 8346 compared to the well-known gap of 8350 following 29370323406802259015...95728858676728143227
sa I have devoted far too many resources to this, I will rest. I also look briefly at the gap following 266190823030249*1129#/210-22844, but the length of time taken to check each possible range of 43k+ is too long. The best I achieved to date is 84 primes following [code]1101306855*1151#+67995358713657430359048762006542336703972224978670437437482633858004501532345946577534465437727848195399060224576423535081766982746433158823827486255141146637104093921266819644253660410020299599441986875748296750154110874438401578094603567430369998521465621565610168020569114152417095857527450304064588327045566434613143149884391737286419623885764232620049541559250548525133540166835094146124824189204240031275094620798491331644219231576586550944407818428480069934923985835440814277.[/code] I found two other multipliers 1101311064 and 1101330536 giving the same 84 prime result. The closeness of the multipliers suggests that 100 primes is quite possible. |
Herr Ober, Zahlen bitte!
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Data for maximal gaps for p < 3*10[SUP]13[/SUP] and k <= 109 is now publicly available! Rejoice!
I'm probably taking this up to p = 10[SUP]14[/SUP]. Well, unless anyone wants to join in. Since the primes at the start of a maximal gap almost always* come in clusters, I did a quick check which p[SUB]n[/SUB] had the highest number of occurrences for k <= 100, for 3*10[SUP]13[/SUP] downwards: [SIZE=1]* I know that may be a rather daring statement...[/SIZE] [CODE]#occ p_n 2 29418557625949 (k = 11, 16) 4 29418557625841 (k = 13, 14, 17, 18) 21 29077945916363 (55 <= k <= 85) 23 1376589410333 (55 <= k <= 87) 30 16025473729 (52 <= k <= 98) 33 3099587 (48 <= k <= 100) 34 18313 (47 <= k <= 95) 39 1621 (24 <= k <= 96) 45 661 (18 <= k <= 100) 52 467 (9 <= k <= 99) 66 283 (6 <= k <= 100) 68 199 (2 <= k <= 96) 73 109 (2 <= k <= 100) 77 7 100 2[/CODE]2 and 3 always occur as primes preceding maximal gaps. 5 doesn't always occur since for p = 3 (technically p[SUB]2[/SUB] = 3), for some k, p[SUB]2+k[/SUB] and p[SUB]2+k+1[/SUB] are twin primes and in that case for p = 5 the gap length is the same as for p = 3. However, whenever 5 doesn't appear as a maximal gap, then 7 definitely does, and with respect to the number of occurrences, 7 is either in the lead by one or ties with 5. No p > 7 appears more often than p = 7 as a prime preceding a maximal gap for k = 1, 2, 3, ..., so p = 7 is a local maximum here. But let's do this more formally: Let [$]p_n[/$] be the set of prime numbers and [$]o_n(x)[/$] the set of the number of occurrences of [$]p_n[/$] as primes preceding a maximal gap for all positive integers [$]k <= x[/$]. [$]p_n = \{2, 3, 5, 7, 11, ...\}[/$] [$]o_n(1) = \{1, 1, 0, 1, 0, 0, 0, 0, 1, 0, ...\}[/$] [$]o_n(1000) = \{1000, 1000, 827, 828, 658, 781, 660, 783, 661, 416, ...\}[/$] [$]o_n[/$] and the corresponding [$]p_n[/$] constitutes a local maximum for the above table - in this case for x = 100 - if there does not exist [$]m > n[/$] such that [$]o_m(x) > o_n(x)[/$]. Conjecture: as [$]k \to \infty[/$], the smallest [$]p_n[/$] in the above table with a local maximum of number of occurrences as maximal gap commencers will be fixed. 19 chimes in for a larger range of [$]k[/$], so the list of local maxima [$]p_n[/$] will probably start {2, 7, 19, 109, 199, 9439 (?), ...} for k sufficiently large - this appears to be [I]very[/I] tricky, at least numerically... A follow-up question will be: for fixed x, at what point will the list of local maxima p[SUB]n[/SUB] be settled? For example, in the above table for x = 100, could there be a larger p[SUB]n[/SUB] preceding a maximal gap for more than half of the values of k (in which case o[SUB]n[/SUB] = 45 / p[SUB]n[/SUB] = 661 and possibly o[SUB]n[/SUB] = 52 / p[SUB]n[/SUB] = 467 will be superseded)? Or could there be a gap between consecutive primes so large that all - or at least most - of the p[SUB]n[/SUB] for k > 1 also turn out as maximal gaps? Once creativity strikes... k = 6 is the first k for which p[SUB]n[/SUB] = 2, 3, 5, and 7 each start a maximal gap. For k = 12, all of the first five primes appear in the attached list. For k = 19, this makes six primes, and the first 13 (!) primes appear at k = 68 (so p[SUB]n+68[/SUB]-p[SUB]n[/SUB] becomes continually larger for every p[SUB]n[/SUB] <= 41). I bet MattcAnderson would like to see this sequence in the OEIS :wink: I guess I'm biting off more than I can chew... :smile: |
[QUOTE=Bobby Jacobs;592658]For each k, what are the first few gaps with record CSG ratio? This is very interesting.[/QUOTE]
These are the current record CSG for each k @ p <= 3.9*10[SUP]13[/SUP]: [CODE]k gap CSG p 1 766 0.8177620175 19581334192423 2 900 0.8918228764 21185697626083 3 986 0.9209295055 21185697625997 4 1034 0.9113778510 21185697625949 5 1080 0.9011654792 21185697625903 6 1154 0.8975282707 30103357357379 7 1148 0.8849957771 14580922576079 8 790 0.9265178066 11878096933 9 1316 0.9531616349 14580922575911 10 726 0.9509666672 866956873 11 754 0.9409492473 866956873 12 784 0.9363085666 866956873 13 1448 0.9564495245 5995661470529 14 1496 0.9574428891 5995661470481 15 1322 0.9535221550 396016668869 16 1358 0.9465344483 396016668833 17 1688 0.9836927546 8281634108801 18 1722 0.9710521630 8281634108767 19 1812 1.0165154301 8281634108677 20 1830 0.9880814955 8281634108677 21 1844 0.9563187743 8281634108663 22 1680 0.9463064905 968269822189 23 1890 0.9406396232 6200995919731 24 2134 0.9570149690 38986211476747 25 1780 0.9686207607 628177622389 26 2014 0.9341035539 6200995919683 27 1846 0.9534113552 628177622323 28 2088 0.9679949599 3999281381923 29 2116 0.9536970232 3999281381923 30 2400 0.9501210087 38029505632477 31 2478 0.9762139574 38986211476403 32 2524 0.9768240786 38986211476357 33 2560 0.9689295531 38986211476321 34 2286 0.9703645150 2481562496471 35 2320 0.9639271592 2481562496437 36 2616 0.9834171539 17931997861517 37 2396 0.9895774988 1933468592177 38 2444 0.9981020350 1933468592129 39 2472 0.9863866064 1933468592101 40 2538 0.9821956613 2481562496219 41 2760 0.9803005126 10631985435829 42 2380 0.9991966853 327076778191 43 2392 0.9719895984 327076778179 44 2442 0.9873916591 327076778129 45 2470 0.9784290501 327076778101 46 2762 0.9706117929 2481562496219 47 2520 0.9545666043 327076778051 48 2776 0.9415708602 1933468592101 49 3038 0.9415271787 10026387088493 50 3092 0.9531007373 10026387088439 51 2946 0.9460969948 2796148447381 52 2976 0.9382202652 2796148447381 53 3196 0.9187382475 11783179421593 54 3224 0.9279160571 10026387088493 55 3278 0.9396521374 10026387088439 56 3096 0.9237957124 2481562495661 57 3390 0.9461117876 11783179421371 58 3560 0.9395747528 29077945916363 59 3594 0.9376826431 28158788983159 60 3636 0.9343561260 29077945916363 61 3654 0.9164223001 29077945916363 62 3456 0.9287125490 5716399254341 63 3294 0.9469610659 1376589410369 64 3330 0.9464086867 1376589410333 65 3596 0.9378033618 6215409275507 66 3678 0.9740832743 6215409275249 67 3702 0.9617861382 6215409275249 68 3758 0.9762827903 6215409275249 69 3854 1.0242911884 6215409275249 70 3870 1.0052760984 6215409275249 71 3920 1.0147688787 6215409275249 72 3932 0.9927489370 6215409275237 73 3966 0.9891020412 6215409275041 74 4062 1.0366412505 6215409275041 75 4078 1.0180858187 6215409275041 76 4128 1.0276414005 6215409275041 77 4150 1.0142622729 6215409275407 78 4200 1.0238470491 6215409275357 79 4308 1.0809994193 6215409275249 80 4328 1.0659029505 6215409275249 81 4340 1.0444870805 6215409275237 82 4380 1.0459795515 6215409275177 83 4414 1.0426566161 6215409275143 84 4516 1.0944353381 6215409275041 85 4536 1.0796801338 6215409275041 86 4548 1.0586702538 6215409275029 87 4556 1.0347395141 6215409275021 88 4578 1.0221867581 6215409275041 89 4596 1.0066376308 6215409275041 90 4620 0.9959600976 6215409275041 91 4642 0.9838544524 6215409275041 92 5020 0.9684580361 36683716323913 93 5058 0.9781413471 33994032583531 94 5146 1.0006726694 36683716323913 95 5194 1.0063137564 36683716323913 96 5278 1.0371216659 36683716324039 97 5404 1.0977245069 36683716323913 98 5418 1.0792569593 36683716323899 99 5470 1.0876676245 36683716323847 100 5482 1.0680270856 36683716323847 101 5526 1.0708730803 36683716323791 102 5590 1.0876834546 36683716323913 103 5638 1.0933231416 36683716323913 104 5656 1.0781126752 36683716323847 105 5704 1.0837889389 36683716323847 106 5758 1.0936239342 36683716323913 107 5772 1.0758527238 36683716323899 108 5824 1.0843154811 36683716323847 109 5830 1.0612869894 36683716323841 Bonus: some instances CSG > 1 for k <= 1024 and p <= 2*10^12: 210 7700 1.0009864925 185067241757 211 7746 1.0126426509 185067241757 212 7760 1.0003343480 185067241757 213 7790 1.0000214554 185067241757[/CODE] |
[QUOTE=mart_r;601471]Data for maximal gaps for p < 3*10[SUP]13[/SUP] and k <= 109 is now publicly available! Rejoice!
I'm probably taking this up to p = 10[SUP]14[/SUP]. Well, unless anyone wants to join in. Since the primes at the start of a maximal gap almost always* come in clusters, I did a quick check which p[SUB]n[/SUB] had the highest number of occurrences for k <= 100, for 3*10[SUP]13[/SUP] downwards: [SIZE=1]* I know that may be a rather daring statement...[/SIZE] [CODE]#occ p_n 2 29418557625949 (k = 11, 16) 4 29418557625841 (k = 13, 14, 17, 18) 21 29077945916363 (55 <= k <= 85) 23 1376589410333 (55 <= k <= 87) 30 16025473729 (52 <= k <= 98) 33 3099587 (48 <= k <= 100) 34 18313 (47 <= k <= 95) 39 1621 (24 <= k <= 96) 45 661 (18 <= k <= 100) 52 467 (9 <= k <= 99) 66 283 (6 <= k <= 100) 68 199 (2 <= k <= 96) 73 109 (2 <= k <= 100) 77 7 100 2[/CODE]2 and 3 always occur as primes preceding maximal gaps. 5 doesn't always occur since for p = 3 (technically p[SUB]2[/SUB] = 3), for some k, p[SUB]2+k[/SUB] and p[SUB]2+k+1[/SUB] are twin primes and in that case for p = 5 the gap length is the same as for p = 3. However, whenever 5 doesn't appear as a maximal gap, then 7 definitely does, and with respect to the number of occurrences, 7 is either in the lead by one or ties with 5. No p > 7 appears more often than p = 7 as a prime preceding a maximal gap for k = 1, 2, 3, ..., so p = 7 is a local maximum here. But let's do this more formally: Let [$]p_n[/$] be the set of prime numbers and [$]o_n(x)[/$] the set of the number of occurrences of [$]p_n[/$] as primes preceding a maximal gap for all positive integers [$]k <= x[/$]. [$]p_n = \{2, 3, 5, 7, 11, ...\}[/$] [$]o_n(1) = \{1, 1, 0, 1, 0, 0, 0, 0, 1, 0, ...\}[/$] [$]o_n(1000) = \{1000, 1000, 827, 828, 658, 781, 660, 783, 661, 416, ...\}[/$] [$]o_n[/$] and the corresponding [$]p_n[/$] constitutes a local maximum for the above table - in this case for x = 100 - if there does not exist [$]m > n[/$] such that [$]o_m(x) > o_n(x)[/$]. Conjecture: as [$]k \to \infty[/$], the smallest [$]p_n[/$] in the above table with a local maximum of number of occurrences as maximal gap commencers will be fixed. 19 chimes in for a larger range of [$]k[/$], so the list of local maxima [$]p_n[/$] will probably start {2, 7, 19, 109, 199, 9439 (?), ...} for k sufficiently large - this appears to be [I]very[/I] tricky, at least numerically... A follow-up question will be: for fixed x, at what point will the list of local maxima p[SUB]n[/SUB] be settled? For example, in the above table for x = 100, could there be a larger p[SUB]n[/SUB] preceding a maximal gap for more than half of the values of k (in which case o[SUB]n[/SUB] = 45 / p[SUB]n[/SUB] = 661 and possibly o[SUB]n[/SUB] = 52 / p[SUB]n[/SUB] = 467 will be superseded)? Or could there be a gap between consecutive primes so large that all - or at least most - of the p[SUB]n[/SUB] for k > 1 also turn out as maximal gaps? Once creativity strikes... k = 6 is the first k for which p[SUB]n[/SUB] = 2, 3, 5, and 7 each start a maximal gap. For k = 12, all of the first five primes appear in the attached list. For k = 19, this makes six primes, and the first 13 (!) primes appear at k = 68 (so p[SUB]n+68[/SUB]-p[SUB]n[/SUB] becomes continually larger for every p[SUB]n[/SUB] <= 41). I bet MattcAnderson would like to see this sequence in the OEIS :wink: I guess I'm biting off more than I can chew... :smile:[/QUOTE] How many times does 1327 appear in the list? 1327 has some big gaps to the next primes (1361, 1367, 1373, 1381, 1399, 1409, 1423). What about 1321? Since 1321 is near 1327, it should also appear a lot. |
1 Attachment(s)
[QUOTE=Bobby Jacobs;602186]How many times does 1327 appear in the list? 1327 has some big gaps to the next primes (1361, 1367, 1373, 1381, 1399, 1409, 1423). What about 1321? Since 1321 is near 1327, it should also appear a lot.[/QUOTE]
You're right. For small x, 1327 and some of the previous primes should occur quite often as primes preceding maximal gaps. For x >= 8, 1321 occurs more often than 1327, and for x >= 10, 1303 or 1307 occur more often than 1321. Here's a list for the first 300 primes and the number of occurrences at x = 1000 (i.e. for all k <= 1000) - you clearly see the patterns juxtaposed to the gaps between the consecutive primes: [CODE] p_n o_n(1000) 2 1000 3 1000 5 827 7 828 11 658 13 781 17 660 19 783 23 661 29 416 31 710 37 408 41 558 43 742 47 658 53 418 59 353 61 687 67 401 71 555 73 741 79 416 83 572 89 406 97 260 101 409 103 664 107 625 109 778 113 669 127 104 131 247 137 254 139 524 149 193 151 433 157 330 163 306 167 497 173 363 179 328 181 653 191 219 193 481 197 568 199 745 211 161 223 84 227 199 229 372 233 476 239 352 241 622 251 216 257 272 263 269 269 285 271 572 277 373 281 541 283 731 293 238 307 76 311 184 313 370 317 470 331 93 337 144 347 90 349 278 353 375 359 304 367 218 373 248 379 258 383 414 389 333 397 239 401 393 409 241 419 144 421 374 431 170 433 409 439 316 443 484 449 368 457 250 461 407 463 667 467 627 479 163 487 159 491 298 499 208 503 345 509 306 521 114 523 353 541 37 547 80 557 60 563 104 569 128 571 296 577 233 587 135 593 179 599 204 601 450 607 308 613 291 617 472 619 667 631 156 641 121 643 317 647 428 653 354 659 320 661 628 673 157 677 328 683 297 691 224 701 142 709 135 719 94 727 106 733 143 739 174 743 303 751 190 757 228 761 373 769 230 773 369 787 88 797 74 809 47 811 158 821 90 823 242 827 332 829 529 839 200 853 65 857 167 859 344 863 431 877 94 881 218 883 445 887 493 907 39 911 115 919 95 929 76 937 90 941 178 947 182 953 197 967 68 971 157 977 175 983 204 991 177 997 206 1009 87 1013 196 1019 208 1021 449 1031 182 1033 404 1039 310 1049 187 1051 416 1061 202 1063 434 1069 335 1087 47 1091 146 1093 314 1097 418 1103 342 1109 325 1117 249 1123 274 1129 285 1151 27 1153 97 1163 76 1171 82 1181 65 1187 96 1193 126 1201 116 1213 58 1217 143 1223 156 1229 185 1231 414 1237 291 1249 112 1259 92 1277 15 1279 70 1283 159 1289 167 1291 352 1297 271 1301 411 1303 600 1307 580 1319 164 1321 424 1327 335 1361 0 1367 7 1373 23 1381 22 1399 2 1409 3 1423 1 1427 9 1429 35 1433 64 1439 54 1447 44 1451 107 1453 227 1459 183 1471 71 1481 58 1483 177 1487 283 1489 439 1493 467 1499 336 1511 123 1523 63 1531 84 1543 49 1549 77 1553 167 1559 172 1567 151 1571 270 1579 190 1583 311 1597 85 1601 190 1607 211 1609 453 1613 486 1619 370 1621 657 1627 406 1637 226 1657 27 1663 59 1667 132 1669 303 1693 11 1697 49 1699 131 1709 89 1721 47 1723 147 1733 83 1741 94 1747 126 1753 158 1759 180 1777 24 1783 57 1787 136 1789 290 1801 99 1811 82 1823 45 1831 54 1847 16 1861 7 1867 18 1871 49 1873 113 1877 174 1879 301 1889 143 1901 75 1907 116 1913 143 1931 23 1933 102 1949 24 1951 93 1973 6 1979 17 1987 21 [/CODE]As one might expect, 1361 has 0 occurrences (the next prime with 0 occurrences for x = 1000 is 2203). (Note also that 1621 occurs more often than 1303. This is mostly because there are rather many primes between 1400 and 1500 but rather few between 1700 and 1800 as well as between 1800 and 1900.) The first time p[SUB]218[/SUB] = 1361 appears as a prime preceding a maximal gap is for k = 1315 because p[SUB]217+1315[/SUB] = p[SUB]1532[/SUB] = 12853 and p[SUB]218+1315[/SUB] = p[SUB]1533[/SUB] = 12889, which is a gap of 36 between consecutive primes (i.e. more than the 34 between 1327 and 1361) and a gap of 11528 between p[SUB]218[/SUB] and p[SUB]1533[/SUB], while for all n < 218, p[SUB]n+1315[/SUB]-p[SUB]n[/SUB] < 11528. If you'd like to play around with a larger set of data, check out the attachment.:smile: |
[QUOTE=mart_r;601471]
Conjecture: as [$]k \to \infty[/$], the smallest [$]p_n[/$] in the above table with a local maximum of number of occurrences as maximal gap commencers will be fixed. 19 chimes in for a larger range of [$]k[/$], so the list of local maxima [$]p_n[/$] will probably start {2, 7, 19, 109, 199, 9439 (?), ...} for k sufficiently large - this appears to be [I]very[/I] tricky, at least numerically... [/QUOTE] I believe that as [$]n\to\infty[/$], the primes p with the most occurrences will be based upon a lot of small prime gaps immediately before p. Therefore, 5659 should eventually beat 109 because the 5 prime gaps before 5659 are 6, 4, 2, 4, 2, but the 5 prime gaps before 109 are 8, 4, 2, 4, 2. |
1 Attachment(s)
[QUOTE=Bobby Jacobs;602880]I believe that as [$]n\to\infty[/$], the primes p with the most occurrences will be based upon a lot of small prime gaps immediately before p. Therefore, 5659 should eventually beat 109 because the 5 prime gaps before 5659 are 6, 4, 2, 4, 2, but the 5 prime gaps before 109 are 8, 4, 2, 4, 2.[/QUOTE]
p=5659 is not a good candidate for a record number of maximal gaps after p, as you can see in the attached graph. The graph shows p[SUB]n[/SUB] vs. o[SUB]n[/SUB](x) at x=500000. Points further to the right have a higher number of occurrences. 5659 is the 746th prime number. o[SUB]746[/SUB](x)=423464, while for p=9439, we already have o[SUB]1170[/SUB](x)=444555. And, just as an aside, [$]\lim_{x\to\infty} x/o_n(x) = 1[/$] (working out secondary terms will be interesting;). Whether 9439 would eventually beat 109 remains to be seen... |
What do you get if you multiply six by nine?
9439 beats 283 at around x=740000.
9439 does not appear to beat 199. 113173 may be the subsequent local maximum (beating 24109 for some x < 1.2e6). A lot more o[SUB]k[/SUB] and a lot higher bound x would need to be looked at to see whether that remains true. Note that 113173 is the penultimate number of an almost-decuplet or cousin-nonuplet or whatever you may call it. So Bobby's observation holds true at this point, with my addition that some large gaps directly after such a cluster (or, say, (p-[$]\theta[/$](p))/[$]\sqrt{p}[/$] is not "too large", YMMV) make for good conditions to produce such "high performer" initial members of these generalized maximal gaps. We may invoke the performance indicator [$]\lim_{x\to\infty} \frac{x}{(\log x -1)(x-o_n(x))}[/$]. More sophisticated ideas are welcome. In principle it might be possible that there exists a larger p that eventually beats 9439, or even 199 or 109 or...?? Intricate problem, delicate computation. Relocate focus? Allocate more resources? [COLOR="LemonChiffon"]Vindicate my existence??[/COLOR] [CODE] k p_k o_k(1e6) 1 2 1000000 2 3 1000000 3 5 913974 4 7 913975 5 11 828143 6 13 901885 7 17 828145 8 19 901887 9 23 828146 10 29 681628 11 31 886659 12 37 680180 13 41 800714 14 43 896535 15 47 827790 16 53 681558 17 59 658923 18 61 883217 19 67 679222 20 71 800359 21 73 896477 22 79 681232 23 83 801182 24 89 678585 25 97 592056 26 101 752065 27 103 889285 28 107 825738 29 109 901630 30 113 828113 31 127 381766 32 131 641396 33 137 629027 34 139 864356 35 149 532451 36 151 807001 37 157 668832 38 163 655002 39 167 793753 40 173 676703 41 179 657575 42 181 882331 43 191 535553 44 193 808696 45 197 814221 46 199 899274 47 211 440323 48 223 366557 49 227 639491 50 229 823828 51 233 811478 52 239 678221 53 241 884641 54 251 536345 55 257 619570 56 263 637297 57 269 645947 58 271 875046 59 277 678153 60 281 799895 61 283 896333 62 293 538088 63 307 340314 64 311 607258 65 313 814055 66 317 807355 67 331 378875 68 337 535249 69 347 461147 70 349 761814 71 353 795136 72 359 673490 73 367 590485 74 373 629575 75 379 641650 76 383 785544 77 389 674505 78 397 590903 79 401 751480 80 409 594445 81 419 489062 82 421 784242 83 431 519907 84 433 800681 85 439 667121 86 443 795923 87 449 677548 88 457 591710 89 461 751923 90 463 889212 91 467 825736 92 479 440288 93 487 511336 94 491 705113 95 499 578974 96 503 746090 97 509 664344 98 521 421689 99 523 763580 100 541 271180 101 547 450118 102 557 429483 103 563 551413 104 569 601162 105 571 843445 106 577 669076 107 587 517671 108 593 609227 109 599 633704 110 601 866713 111 607 676096 112 613 657193 113 617 794421 114 619 894991 115 631 440101 116 641 449185 117 643 747316 118 647 793640 119 653 673934 120 659 656689 121 661 881502 122 673 438065 123 677 699049 124 683 649989 125 691 579297 126 701 484408 127 709 518248 128 719 461765 129 727 503357 130 733 585160 131 739 619697 132 743 774027 133 751 594239 134 757 631679 135 761 779383 136 769 597441 137 773 756587 138 787 375604 139 797 405533 140 809 343311 141 811 677349 142 821 485015 143 823 773652 144 827 798438 145 829 893732 146 839 537224 147 853 340043 148 857 606855 149 859 813837 150 863 807195 151 877 378877 152 881 638598 153 883 841158 154 887 813505 155 907 244026 156 911 503828 157 919 497066 158 929 443645 159 937 494496 160 941 691894 161 947 644162 162 953 644277 163 967 367061 164 971 621546 165 977 619750 166 983 634068 167 991 581393 168 997 624884 169 1009 412527 170 1013 674085 171 1019 641875 172 1021 866845 173 1031 533531 174 1033 807228 175 1039 668959 176 1049 518833 177 1051 798028 178 1061 526147 179 1063 802249 180 1069 667699 181 1087 271076 182 1091 552183 183 1093 776155 184 1097 796662 185 1103 674267 186 1109 656548 187 1117 588379 188 1123 628691 189 1129 641397 190 1151 206487 191 1153 504779 192 1163 433703 193 1171 476260 194 1181 442801 195 1187 560904 196 1193 605981 197 1201 562730 198 1213 393293 199 1217 655929 200 1223 631357 201 1229 641244 202 1231 871744 203 1237 676947 204 1249 424889 205 1259 442089 206 1277 230001 207 1279 537647 208 1283 688222 209 1289 635527 210 1291 860410 211 1297 673956 212 1301 797723 213 1303 895422 214 1307 827464 215 1319 440624 216 1321 774222 217 1327 659014 218 1361 79813 219 1367 209066 220 1373 340459 221 1381 389501 222 1399 198160 223 1409 279261 224 1423 223083 225 1427 455166 226 1429 679754 227 1433 730388 228 1439 642468 229 1447 572724 230 1451 738132 231 1453 879575 232 1459 676456 233 1471 424651 234 1481 441874 235 1483 737786 236 1487 790351 237 1489 889188 238 1493 826028 239 1499 681223 240 1511 425957 241 1523 358233 242 1531 460726 243 1543 356252 244 1549 512475 245 1553 700593 246 1559 644827 247 1567 578889 248 1571 743381 249 1579 591731 250 1583 752116 251 1597 375099 252 1601 632095 253 1607 625731 254 1609 862911 255 1613 817332 256 1619 679959 257 1621 886123 258 1627 680141 259 1637 521397 260 1657 208206 261 1663 390702 262 1667 604307 263 1669 802036 264 1693 170780 265 1697 423178 266 1699 663931 267 1709 484307 268 1721 374835 269 1723 704772 270 1733 497014 271 1741 517825 272 1747 595137 273 1753 623020 274 1759 639435 275 1777 265861 276 1783 442245 277 1787 669006 278 1789 830933 279 1801 430556 280 1811 444024 281 1823 361014 282 1831 445895 283 1847 273720 284 1861 233467 285 1867 400434 286 1871 610121 287 1873 793294 288 1877 789955 289 1879 885866 290 1889 535359 291 1901 396111 292 1907 544980 293 1913 602869 294 1931 258459 295 1933 599789 296 1949 296307 297 1951 619466 298 1973 188985 299 1979 364879 300 1987 421153 301 1993 519458 302 1997 705400 303 1999 849964 304 2003 811377 305 2011 600373 306 2017 633940 307 2027 506154 308 2029 791754 309 2039 524056 310 2053 335861 311 2063 383029 312 2069 517738 313 2081 376196 314 2083 709380 315 2087 768481 316 2089 882615 317 2099 534823 318 2111 396210 319 2113 730259 320 2129 325708 321 2131 650994 322 2137 619950 323 2141 773393 324 2143 885306 325 2153 536756 326 2161 539390 327 2179 248779 328 2203 103983 329 2207 301353 330 2213 407594 331 2221 435477 332 2237 261184 333 2239 564008 334 2243 688647 335 2251 556641 336 2267 304289 337 2269 627988 338 2273 733979 339 2281 578974 340 2287 623886 341 2293 637550 342 2297 783865 343 2309 435095 344 2311 769225 345 2333 208722 346 2339 389671 347 2341 664115 348 2347 616289 349 2351 767295 350 2357 667379 351 2371 369003 352 2377 528523 353 2381 714310 354 2383 868899 355 2389 675807 356 2393 798560 357 2399 677930 358 2411 425375 359 2417 563845 360 2423 612280 361 2437 360831 362 2441 614207 363 2447 616654 364 2459 408719 365 2467 494829 366 2473 582726 367 2477 744373 368 2503 149821 369 2521 122824 370 2531 198653 371 2539 286461 372 2543 482322 373 2549 516578 374 2551 763367 375 2557 633623 376 2579 202834 377 2591 238824 378 2593 527746 379 2609 275415 380 2617 373286 381 2621 601298 382 2633 384112 383 2647 293422 384 2657 345619 385 2659 651972 386 2663 729281 387 2671 576743 388 2677 619995 389 2683 635299 390 2687 781478 391 2689 889819 392 2693 825487 393 2699 680961 394 2707 592895 395 2711 752225 396 2713 889327 397 2719 679804 398 2729 521338 399 2731 800345 400 2741 526548 401 2749 533833 402 2753 727666 403 2767 369050 404 2777 402203 405 2789 341725 406 2791 673881 407 2797 619118 408 2801 768669 409 2803 884785 410 2819 348263 411 2833 267563 412 2837 536535 413 2843 576410 414 2851 544850 415 2857 604937 416 2861 762727 417 2879 276501 418 2887 403023 419 2897 398584 420 2903 524835 421 2909 587106 422 2917 555130 423 2927 470360 424 2939 368898 425 2953 291244 426 2957 539196 427 2963 578023 428 2969 609911 429 2971 853615 430 2999 135663 431 3001 377224 432 3011 375081 433 3019 435537 434 3023 646243 435 3037 344138 436 3041 600602 437 3049 540405 438 3061 384483 439 3067 534120 440 3079 380813 441 3083 646399 442 3089 628391 443 3109 229125 444 3119 298909 445 3121 609687 446 3137 296193 447 3163 96427 448 3167 273277 449 3169 522994 450 3181 337380 451 3187 485383 452 3191 677429 453 3203 404609 454 3209 544594 455 3217 538341 456 3221 712941 457 3229 581473 458 3251 191377 459 3253 493381 460 3257 662046 461 3259 827402 462 3271 428970 463 3299 110859 464 3301 328491 465 3307 439934 466 3313 523151 467 3319 577109 468 3323 744830 469 3329 657443 470 3331 871040 471 3343 436091 472 3347 697601 473 3359 419318 474 3361 749465 475 3371 511805 476 3373 794333 477 3389 339942 478 3391 663593 479 3407 317447 480 3413 485373 481 3433 196669 482 3449 177667 483 3457 283750 484 3461 496750 485 3463 710739 486 3467 748381 487 3469 864755 488 3491 216373 489 3499 325129 490 3511 295169 491 3517 454675 492 3527 426576 493 3529 706953 494 3533 768968 495 3539 662261 496 3541 874313 497 3547 677090 498 3557 520352 499 3559 799201 500 3571 429423 501 3581 444069 502 3583 742622 503 3593 511697 504 3607 331423 505 3613 490540 506 3617 698211 507 3623 643396 508 3631 578801 509 3637 623427 510 3643 638642 511 3659 328517 512 3671 316903 513 3673 628232 514 3677 738302 515 3691 365020 516 3697 523418 517 3701 714872 518 3709 576977 519 3719 482730 520 3727 516720 521 3733 594282 522 3739 624150 523 3761 203934 524 3767 384383 525 3769 656480 526 3779 481453 527 3793 319156 528 3797 583260 529 3803 599607 530 3821 256905 531 3823 594116 532 3833 455338 533 3847 314010 534 3851 573882 535 3853 792200 536 3863 514446 537 3877 333267 538 3881 598369 539 3889 539793 540 3907 246959 541 3911 510676 542 3917 568363 543 3919 813421 544 3923 802319 545 3929 675323 546 3931 883218 547 3943 438508 548 3947 699625 549 3967 230279 550 3989 120105 551 4001 167646 552 4003 414702 553 4007 561937 554 4013 564455 555 4019 595016 556 4021 835490 557 4027 663485 558 4049 209084 559 4051 506397 560 4057 548296 561 4073 305295 562 4079 473101 563 4091 359147 564 4093 684769 565 4099 623089 566 4111 409079 567 4127 276614 568 4129 583433 569 4133 719779 570 4139 650097 571 4153 364128 572 4157 622098 573 4159 832491 574 4177 277476 575 4201 111320 576 4211 199920 577 4217 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640 4751 268043 641 4759 395347 642 4783 129670 643 4787 342164 644 4789 592210 645 4793 689211 646 4799 631082 647 4801 854264 648 4813 432619 649 4817 693423 650 4831 356979 651 4861 78321 652 4871 157711 653 4877 284942 654 4889 253467 655 4903 216658 656 4909 363386 657 4919 355502 658 4931 304835 659 4933 610047 660 4937 699828 661 4943 629358 662 4951 566710 663 4957 613674 664 4967 495362 665 4969 782624 666 4973 801797 667 4987 377781 668 4993 534444 669 4999 599716 670 5003 756510 671 5009 666924 672 5011 880114 673 5021 535484 674 5023 808415 675 5039 342864 676 5051 324946 677 5059 420624 678 5077 216155 679 5081 467518 680 5087 537471 681 5099 377278 682 5101 706380 683 5107 631773 684 5113 637824 685 5119 643508 686 5147 129819 687 5153 287294 688 5167 225853 689 5171 461103 690 5179 451590 691 5189 417937 692 5197 468058 693 5209 352977 694 5227 202754 695 5231 454876 696 5233 696563 697 5237 752628 698 5261 170833 699 5273 215548 700 5279 365958 701 5281 649499 702 5297 296749 703 5303 461695 704 5309 551476 705 5323 333348 706 5333 374450 707 5347 283052 708 5351 529725 709 5381 86656 710 5387 222602 711 5393 345814 712 5399 454542 713 5407 473749 714 5413 550162 715 5417 715617 716 5419 854932 717 5431 431440 718 5437 566155 719 5441 746288 720 5443 875900 721 5449 676516 722 5471 211813 723 5477 393178 724 5479 665294 725 5483 750823 726 5501 272919 727 5503 610455 728 5507 726831 729 5519 420942 730 5521 756797 731 5527 652803 732 5531 788054 733 5557 152766 734 5563 315357 735 5569 450930 736 5573 639919 737 5581 546089 738 5591 464987 739 5623 75406 740 5639 94364 741 5641 279633 742 5647 369528 743 5651 549308 744 5653 738664 745 5657 754116 746 5659 866159 747 5669 529323 748 5683 337199 749 5689 494898 750 5693 700771 751 5701 569625 752 5711 480614 753 5717 588250 754 5737 222931 755 5741 472373 756 5743 735088 757 5749 636933 758 5779 98556 759 5783 303754 760 5791 356671 761 5801 370129 762 5807 498805 763 5813 562457 764 5821 535155 765 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450680 953 7523 562536 954 7529 604751 955 7537 560954 956 7541 731018 957 7547 657245 958 7549 874039 959 7559 533420 960 7561 806029 961 7573 431111 962 7577 693945 963 7583 647800 964 7589 648250 965 7591 874560 966 7603 437384 967 7607 698495 968 7621 358362 969 7639 209942 970 7643 462514 971 7649 532784 972 7669 209623 973 7673 449842 974 7681 465751 975 7687 557140 976 7691 723945 977 7699 578417 978 7703 742997 979 7717 372437 980 7723 529536 981 7727 718094 982 7741 366036 983 7753 335167 984 7757 586816 985 7759 805664 986 7789 101290 987 7793 310708 988 7817 107529 989 7823 238258 990 7829 362089 991 7841 294758 992 7853 276436 993 7867 237917 994 7873 388145 995 7877 587133 996 7879 770904 997 7883 770281 998 7901 273698 999 7907 450006 1000 7919 350255 1001 7927 447230 1002 7933 548613 1003 7937 718802 1004 7949 417802 1005 7951 753233 1006 7963 419300 1007 7993 83831 1008 8009 107456 1009 8011 299059 1010 8017 386665 1011 8039 149072 1012 8053 154876 1013 8059 291640 1014 8069 314601 1015 8081 278252 1016 8087 422275 1017 8089 677719 1018 8093 725639 1019 8101 564015 1020 8111 474186 1021 8117 580411 1022 8123 614559 1023 8147 162735 1024 8161 184935 1025 8167 335236 1026 8171 539215 1027 8179 495718 1028 8191 362521 1029 8209 204301 1030 8219 291170 1031 8221 567386 1032 8231 444250 1033 8233 727774 1034 8237 773312 1035 8243 663068 1036 8263 235989 1037 8269 417567 1038 8273 619215 1039 8287 342851 1040 8291 598860 1041 8293 814501 1042 8297 801132 1043 8311 378146 1044 8317 534439 1045 8329 383350 1046 8353 134075 1047 8363 224221 1048 8369 368955 1049 8377 412013 1050 8387 394147 1051 8389 680917 1052 8419 93879 1053 8423 293000 1054 8429 408254 1055 8431 662652 1056 8443 385953 1057 8447 646304 1058 8461 341841 1059 8467 504484 1060 8501 69186 1061 8513 116949 1062 8521 202233 1063 8527 315343 1064 8537 321914 1065 8539 585546 1066 8543 680144 1067 8563 219319 1068 8573 287906 1069 8581 387302 1070 8597 244204 1071 8599 542087 1072 8609 433592 1073 8623 297926 1074 8627 552710 1075 8629 770476 1076 8641 415269 1077 8647 554279 1078 8663 306473 1079 8669 475503 1080 8677 489320 1081 8681 691717 1082 8689 567613 1083 8693 738977 1084 8699 661015 1085 8707 586511 1086 8713 627340 1087 8719 640503 1088 8731 417157 1089 8737 558055 1090 8741 744102 1091 8747 663209 1092 8753 652517 1093 8761 586277 1094 8779 258670 1095 8783 523790 1096 8803 200263 1097 8807 446712 1098 8819 335205 1099 8821 658715 1100 8831 475262 1101 8837 579208 1102 8839 828177 1103 8849 524644 1104 8861 391986 1105 8863 725742 1106 8867 781045 1107 8887 240885 1108 8893 422612 1109 8923 80478 1110 8929 208909 1111 8933 409046 1112 8941 416152 1113 8951 393585 1114 8963 325890 1115 8969 479337 1116 8971 747736 1117 8999 127243 1118 9001 360626 1119 9007 464505 1120 9011 660889 1121 9013 818627 1122 9029 337805 1123 9041 320627 1124 9043 629664 1125 9049 607921 1126 9059 494063 1127 9067 516754 1128 9091 150623 1129 9103 198792 1130 9109 346340 1131 9127 185103 1132 9133 341617 1133 9137 558076 1134 9151 310334 1135 9157 467711 1136 9161 665337 1137 9173 401783 1138 9181 480776 1139 9187 571163 1140 9199 393381 1141 9203 652871 1142 9209 630918 1143 9221 411875 1144 9227 553895 1145 9239 387908 1146 9241 731940 1147 9257 323508 1148 9277 166449 1149 9281 398003 1150 9283 651804 1151 9293 470813 1152 9311 234415 1153 9319 358320 1154 9323 573906 1155 9337 328114 1156 9341 581934 1157 9343 798460 1158 9349 656105 1159 9371 207995 1160 9377 388085 1161 9391 271152 1162 9397 437220 1163 9403 530335 1164 9413 458656 1165 9419 567130 1166 9421 815835 1167 9431 522055 1168 9433 796818 1169 9437 809480 1170 9439 896807 1171 9461 220160 1172 9463 526660 1173 9467 691318 1174 9473 640024 1175 9479 642888 1176 9491 417162 1177 9497 557734 1178 9511 344895 1179 9521 380990 1180 9533 331294 1181 9539 485629 1182 9547 508534 1183 9551 691160 1184 9587 63385 1185 9601 95979 1186 9613 130162 1187 9619 247563 1188 9623 418077 1189 9629 464520 1190 9631 708552 1191 9643 389335 1192 9649 529232 1193 9661 374089 1194 9677 261811 1195 9679 562329 1196 9689 453557 1197 9697 488792 1198 9719 176513 1199 9721 459824 1200 9733 333283 1201 9739 482070 1202 9743 683872 1203 9749 633336 1204 9767 263156 1205 9769 602146 1206 9781 381785 1207 9787 531735 1208 9791 718107 1209 9803 419918 1210 9811 495878 1211 9817 583689 1212 9829 399246 1213 9833 659599 1214 9839 635254 1215 9851 413756 1216 9857 555795 1217 9859 826057 1218 9871 429932 1219 9883 362286 1220 9887 633945 1221 9901 341356 1222 9907 506371 1223 9923 290652 1224 9929 457519 1225 9931 736097 1226 9941 501827 1227 9949 519633 1228 9967 243664 1229 9973 418661 1230 10007 63093 1231 10009 226219 1232 10037 64385 1233 10039 212523 1234 10061 96217 1235 10067 212622 (...) 2684 24109 889952 :727 113173 889409 [/CODE] For these k, the first n primes are preceding generalized maximal gaps p[SUB]n+k[/SUB]-p[SUB]n[/SUB]: [CODE] n k 2 1 3 2 4 6 5 12 6 19 7 97 8 70 9 120 10 88 11 119 12 237 13 68 14 681 15 412 16 1591 17 2907 18 1510 19 2734 20 2131 21 1588 22 3834 23 6041 24 2897 25 11562 26 21004 27 11560 28 44194 29 21001 30 11557 31 25174 32 32114 33 131271 34 36918 35 44636 36 115242 37 211442 38 477957 39 64935 40 204412 41 710665 42 175930 43 438049 44 409641 45 725804 46 176350 47 560510 48 2570641 49 2841381 50 4094784 51 1063896 52 4355669 53 1807346 54 2070798 55 2349691 56 6380527 57 6563887 58 6276812 59 14215737 60 8543349 61 2899899 62 7714640 63 19264207 64 15644556 65 13668980 66 10701209 67 24451150 68 13668996 69 38417236 70 33907310 71 25958214 72 37376935 73 72210305 74 51624533 75 155807588 76 121101282 77 72019160 78 199395703 79 34335444 80 80104183 81 575130837 82 273221126 83 362546538 84 478749161 85 209832527 86 92967699 87 251653222 90 833367050 91 566487675 92 212341969 93 838711510 94 394795699 97 457331290 99 864115614 107 834990586 Search limit: k=9e8 [/CODE] And now for the cherry on top of it: For 25698372294281 <= p <= 25698372297167 there are 144 values of k with 302 <= k <= 445 for which a new CSG maximum is > 1, with the largest instance at p = 25698372297029, k = 316, CSG = 1.09729237... Ah, the fun we have :smile: |
[QUOTE=mart_r;604557]Relocate focus?[/QUOTE]
That's what. You know, even though I don't get many replies, it helps that I share my ideas here as it puts more pressure on me to think things through more thoroughly (try saying that five times fast:), beneath all my rampant numerology. [QUOTE=Bobby Jacobs;602880]I believe that as [$]n\to\infty[/$], the primes p with the most occurrences will be based upon a lot of small prime gaps immediately before p. Therefore, 5659 should eventually beat 109 because the 5 prime gaps before 5659 are 6, 4, 2, 4, 2, but the 5 prime gaps before 109 are 8, 4, 2, 4, 2.[/QUOTE] That seems to be right after all - I stand corrected. Those "high performer" primes preceding maximal gaps depend primarily on the small gaps right before them. I can see it now - it might be well out of reach for an actual computation, but on an asymptotic scale, 5659, being the last member of a prime-septuplet, does have a good chance to beat 109 sometime. |
What is the pattern with the sequence of primes with record low numbers of occurrences? It seems like the sequence is 2, 5, 11, 29, 37, 59, 97, 127, 223, 307, 541, 907, 1151, 1361, ... This is similar to the primes at the end of maximal prime gaps, but not exactly. I wonder what the pattern is.
|
Me too :smile:
At first sight, 37 should occur more often than 29 because the two gaps preceding 37 are {2, 6} instead of {4, 6} for 29. If however we take three gaps before the prime into account, it's {6, 2, 6} vs. {2, 4, 6}. The {2, 4, 6}-pattern having more open residues mod 5 also plays a role, favoring 37 as a local record minimum in number of occurrences. Now, at what margin remains 37 below 29? |
Forbidden prime gap combinations
Let an n-prime gap be the gap between a prime p and the prime n primes after p. Then, 2 and 3 are always the start of a maximal n-gap for all n. 5 is the start of a maximal n-gap if and only if the (n+2)nd prime and the (n+3)rd prime are not twin primes. 7 is the start of a maximal n-gap if and only if the (n+3)rd prime and the (n+4)th prime are not twin primes. 11 is the start of a maximal n-gap if and only if the (n+4)th and (n+5)th primes have a gap greater than 4. 13 is the start of a maximal n-gap if and only if the (n+5)th and (n+6)th primes are not twin primes, and the last 2 gaps before the (n+6)th prime are not (2, 4). Basically, every prime has a set of "forbidden prime gap combinations" such that the mth prime is the start of a maximal n-gap if and only if the last gaps before the (m+n)th prime are not one of the forbidden gap combinations. Here are the forbidden gap combinations of the first few primes.
[CODE] 2 [] 3 [] 5 [[2]] 7 [[2]] 11 [[2], [4]] 13 [[2], [2, 4]] 17 [[2], [4]] 19 [[2], [2, 4]] 23 [[2], [4]] 29 [[2], [4], [6]] 31 [[2], [2, 4], [2, 6], [2, 6, 4], [2, 4, 6]] 37 [[2], [4], [6], [2, 4, 8], [2, 4, 2, 10]] 41 [[2], [4], [2, 6], [4, 6], [2, 4, 6, 6], [4, 2, 4, 8], [2, 4, 2, 10]] 43 [[2], [2, 4], [2, 6, 4], [4, 2, 6], [2, 4, 6], [2, 4, 6, 2, 6], [2, 4, 2, 4, 8]] 47 [[2], [4], [4, 2, 4, 6], [4, 2, 4, 6, 2, 6], [4, 2, 4, 2, 4, 8]] [/CODE] Notice that 29 just has the forbidden gaps 2, 4, 6, but 37 has the extra combinations (2, 4, 8) and (2, 4, 2, 10). That is why 29 is more common than 37. |
[QUOTE=Bobby Jacobs;604986]Here are the forbidden gap combinations of the first few primes.
[/QUOTE] Very good! That's the sort of analysis I was looking for. Do you have a program for these gap combinations? |
Yes. I have a program, but it is slow for primes above 47. We basically want admissible k-tuples where the total of the gaps is less than or equal to the total of the k gaps before p. Let p[SUB]m[/SUB] be the mth prime. Suppose the k gaps before the (m+n)th prime are one of these forbidden k-tuples. If p[SUB]m+n[/SUB]-p[SUB]m+n-k[/SUB]<=p[SUB]m[/SUB]-p[SUB]m-k[/SUB], then p[SUB]m+n-k[/SUB]-p[SUB]m-k[/SUB]>=p[SUB]m+n[/SUB]-p[SUB]m[/SUB]. Then, the (m-k)th prime will have at least as big of an n-gap as the mth prime. Therefore, the forbidden gaps are minimal admissible k-tuples >= the k gaps before p[SUB]m[/SUB].
|
CSG[SUB]max[/SUB] for p<=10[SUP]14[/SUP]:
[CODE] k gap CSG_max p 1 766 0.81776202 19581334192423 2 900 0.89182288 21185697626083 3 986 0.92092951 21185697625997 4 1134 0.93874248 66592576389587 5 1170 0.91718026 66592576389551 6 1154 0.89752827 30103357357379 7 1148 0.88499578 14580922576079 8 790 0.92651781 11878096933 9 1316 0.95316163 14580922575911 10 726 0.95096666 866956873 11 754 0.94094924 866956873 12 784 0.93630856 866956873 13 1448 0.95644952 5995661470529 14 1496 0.95744289 5995661470481 15 1322 0.95352216 396016668869 16 1358 0.94653445 396016668833 17 1688 0.98369275 8281634108801 18 1722 0.97105216 8281634108767 19 1812 1.01651543 8281634108677 20 1830 0.98808150 8281634108677 21 2134 1.02168813 78736011999913 22 2148 0.99072269 78736011999913 23 2166 0.96394446 78736011999913 24 2310 1.04764008 78736011999913 25 2322 1.01591301 78736011999901 26 2338 0.98829568 78736011999913 27 2376 0.98009540 78736011999847 28 2432 0.98752862 78736011999791 29 2454 0.96623635 78736011999769 30 2494 0.96053115 78736011999913 31 2478 0.97621396 38986211476403 32 2524 0.97682408 38986211476357 33 2560 0.96892955 38986211476321 34 2286 0.97036452 2481562496471 35 2320 0.96392716 2481562496437 36 2616 0.98341715 17931997861517 37 2396 0.98957750 1933468592177 38 2444 0.99810203 1933468592129 39 2472 0.98638661 1933468592101 40 2538 0.98219566 2481562496219 41 2760 0.98030051 10631985435829 42 2380 0.99919669 327076778191 43 2392 0.97198960 327076778179 44 2442 0.98739166 327076778129 45 2470 0.97842905 327076778101 46 2762 0.97061179 2481562496219 47 2520 0.95456660 327076778051 48 2776 0.94157086 1933468592101 49 3038 0.94152718 10026387088493 50 3092 0.95310074 10026387088439 51 2946 0.94609699 2796148447381 52 2976 0.93822027 2796148447381 53 3450 0.93208471 60681682061173 54 3224 0.92791606 10026387088493 55 3278 0.93965214 10026387088439 56 3096 0.92379571 2481562495661 57 3390 0.94611179 11783179421371 58 3560 0.93957475 29077945916363 59 3808 0.96141677 90210824580841 60 3764 0.95339422 55956455554739 61 3798 0.94719704 55956455554739 62 3852 0.95602954 55956455554651 63 3942 0.99181087 55956455554561 64 3976 0.98566033 55956455554561 65 4004 1.00012038 45921691543349 66 4020 0.98072956 45921691543333 67 4086 0.99893031 45921691543267 68 4140 1.00814094 45921691543213 69 3854 1.02429119 6215409275249 70 4292 1.05955757 45921691543061 71 4310 1.04178765 45921691543043 72 4332 1.02721666 45921691543061 73 4386 1.03648387 45921691543061 74 4062 1.03664125 6215409275041 75 4078 1.01808582 6215409275041 76 4128 1.02764140 6215409275041 77 4150 1.01426227 6215409275407 78 4200 1.02384705 6215409275357 79 4308 1.08099942 6215409275249 80 4328 1.06590295 6215409275249 81 4340 1.04448708 6215409275237 82 4380 1.04597955 6215409275177 83 4414 1.04265662 6215409275143 84 4516 1.09443534 6215409275041 85 4536 1.07968013 6215409275041 86 4548 1.05867025 6215409275029 87 4556 1.03473951 6215409275021 88 4578 1.02218676 6215409275041 89 4596 1.00663763 6215409275041 90 4620 0.99596010 6215409275041 91 4642 0.98385445 6215409275041 92 5020 0.96845804 36683716323913 93 5058 0.97814135 33994032583531 94 5146 1.00067267 36683716323913 95 5194 1.00631376 36683716323913 96 5278 1.03712167 36683716324039 97 5404 1.09772451 36683716323913 98 5418 1.07925696 36683716323899 99 5470 1.08766762 36683716323847 100 5482 1.06802709 36683716323847 101 5526 1.07087308 36683716323791 102 5590 1.08768345 36683716323913 103 5638 1.09332314 36683716323913 104 5656 1.07811268 36683716323847 105 5704 1.08378894 36683716323847 106 5758 1.09362393 36683716323913 107 5772 1.07585272 36683716323899 108 5824 1.08431548 36683716323847 109 5830 1.06128699 36683716323841 [/CODE] And just above 10[SUP]14[/SUP], these 22 new records showed up: [CODE] 10 1528 0.96314466 102591551174059 11 1560 0.94298881 102591551174027 50 3450 0.97333053 102267713449991 51 3480 0.96260938 102267713449991 52 3562 0.99122668 102267713449879 53 3592 0.98063297 102267713449879 54 3634 0.97918812 102267713449807 55 3684 0.98379105 102267713449757 56 3714 0.97357591 102267713449757 57 3768 0.98125523 102267713449673 58 3798 0.97126377 102267713449673 59 3834 0.96582204 102267713449607 60 3874 0.96340363 102267713449567 61 3904 0.95381038 102267713449567 62 3958 0.96169379 102267713449483 66 4186 1.00199403 102267713449117 68 4324 1.03945196 102267713449117 69 4354 1.03013486 102267713449117 76 4658 1.03478754 101562452774609 77 4694 1.03029216 101562452774609 92 5304 1.01634058 102267713449117 93 5328 1.00471893 102267713449093 [/CODE] @ Bobby: I'm working on a program to look for the forbidden gap combinations. If it works, it should be fast enough for primes up to at least 97 (well at least I hope so). |
It appears my VBA code for "forbidden gap combinations" (for getting a heuristic grip on the generalized maximal gap candidates) works as it should:
[CODE] 5: [ 2] 7: [ 2] 11: [ 2], [ 4] 13: [ 2], [ 2, 4] 17: [ 2], [ 4] 19: [ 2], [ 2, 4] 23: [ 2], [ 4] 29: [ 2], [ 4], [ 6] 31: [ 2], [ 2, 4], [ 2, 6], [ 2, 6, 4], [ 2, 4, 6] 37: [ 2], [ 4], [ 6], [ 2, 4, 8], [ 2, 4, 2, 10] 41: [ 2], [ 4], [ 2, 6], [ 4, 6], [ 2, 4, 6, 6], [ 4, 2, 4, 8], [ 2, 4, 2, 10] 43: [ 2], [ 2, 4], [ 2, 6, 4], [ 4, 2, 6], [ 2, 4, 6], [ 2, 4, 6, 2, 6], [ 2, 4, 2, 4, 8] 47: [ 2], [ 4], [ 4, 2, 4, 6], [ 4, 2, 4, 6, 2, 6], [ 4, 2, 4, 2, 4, 8] 53: [ 2], [ 4], [ 6], [ 2, 4, 2, 4, 8], [ 4, 2, 4, 2, 10], [ 2, 4, 2, 4, 6, 2, 10] 59: [ 2], [ 4], [ 6], [ 4, 8], [ 2, 10], [ 2, 4, 2, 4, 6, 8], [ 2, 4, 2, 4, 6, 2, 6, 10] 61: [ 2], [ 2, 4], [ 2, 6], [ 2, 6, 4], [ 2, 4, 6], [ 2, 6, 6], [ 2, 4, 8], [ 2, 6, 6, 4], [ 2, 6, 4, 6], [ 2, 4, 6, 6], [ 2, 4, 2, 10], [ 2, 4, 2, 4, 8, 6, 4], [ 2, 4, 2, 4, 6, 8, 4], [ 4, 2, 4, 2, 4, 8, 6], [ 2, 4, 2, 4, 6, 2, 10], [ 2, 4, 2, 4, 6, 2, 6, 6, 4, 6], [ 2, 4, 2, 4, 6, 2, 6, 4, 6, 6], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 10] 67: [ 2], [ 4], [ 6], [ 2, 4, 8], [ 2, 6, 4, 8], [ 2, 4, 6, 8], [ 2, 4, 2, 10], [ 2, 4, 2, 12], [ 2, 6, 4, 2, 10], [ 2, 4, 6, 2, 10], [ 2, 4, 2, 4, 6, 2, 6, 10], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 12] 71: [ 2], [ 4], [ 2, 6], [ 4, 6], [ 2, 4, 6, 6], [ 4, 2, 4, 8], [ 2, 4, 2, 10], [ 4, 6, 2, 6, 6], [ 2, 6, 4, 6, 6], [ 4, 2, 4, 8, 6], [ 4, 6, 2, 4, 8], [ 4, 2, 6, 4, 8], [ 4, 2, 4, 6, 8], [ 2, 6, 4, 2, 10], [ 2, 4, 6, 2, 10], [ 4, 2, 4, 2, 12], [ 2, 4, 2, 4, 6, 2, 6, 6, 4, 6, 6], [ 2, 4, 2, 4, 6, 2, 6, 4, 6, 6, 6], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 10, 6], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 12] 73: [ 2], [ 2, 4], [ 2, 6, 4], [ 4, 2, 6], [ 2, 4, 6], [ 2, 4, 6, 2, 6], [ 2, 4, 2, 4, 8], [ 2, 6, 4, 6, 2, 6], [ 2, 4, 6, 6, 2, 6], [ 2, 4, 2, 10, 2, 6], [ 2, 4, 6, 2, 6, 6], [ 2, 4, 2, 4, 8, 6], [ 2, 4, 2, 4, 6, 8], [ 2, 4, 6, 2, 6, 6, 4], [ 2, 4, 2, 4, 8, 6, 4], [ 2, 4, 2, 4, 6, 8, 4], [ 2, 6, 4, 2, 6, 4, 6], [ 2, 4, 6, 2, 6, 4, 6], [ 2, 6, 4, 2, 4, 6, 6], [ 2, 6, 4, 2, 4, 2, 10], [ 2, 4, 2, 4, 6, 2, 10], [ 2, 4, 2, 4, 6, 2, 6, 6, 4, 6], [ 2, 4, 2, 4, 6, 2, 6, 4, 6, 6], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 10], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 8] 79: [ 2], [ 4], [ 6], [ 4, 2, 4, 8], [ 2, 4, 2, 10], [ 2, 4, 2, 4, 6, 8], [ 2, 4, 6, 2, 6, 4, 8], [ 2, 6, 4, 2, 4, 6, 8], [ 2, 4, 2, 4, 6, 2, 10], [ 2, 6, 4, 2, 4, 2, 12], [ 2, 4, 2, 4, 6, 2, 12], [ 2, 4, 6, 2, 6, 4, 2, 10], [ 2, 4, 2, 4, 6, 2, 6, 10], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 12], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 8] 83: [ 2], [ 4], [ 2, 6], [ 4, 6], [ 4, 2, 4, 6, 6], [ 2, 4, 2, 4, 8], [ 4, 2, 4, 2, 10], [ 4, 2, 4, 6, 2, 6, 6], [ 4, 2, 4, 2, 4, 8, 6], [ 2, 4, 2, 4, 6, 2, 10], [ 4, 6, 2, 4, 6, 2, 6, 6], [ 4, 2, 4, 6, 6, 2, 6, 6], [ 2, 4, 6, 2, 6, 4, 6, 6], [ 4, 2, 4, 6, 2, 6, 6, 6], [ 2, 4, 2, 4, 6, 2, 10, 6], [ 4, 2, 4, 6, 2, 6, 4, 8], [ 2, 4, 6, 2, 6, 4, 2, 10], [ 2, 4, 2, 4, 6, 2, 6, 10], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 8] 89: [ 2], [ 4], [ 6], [ 2, 4, 8], [ 4, 2, 10], [ 2, 4, 2, 4, 6, 8], [ 4, 2, 4, 6, 2, 10], [ 4, 2, 4, 6, 2, 6, 4, 8], [ 2, 4, 2, 4, 6, 2, 6, 10], [ 4, 2, 4, 6, 6, 2, 6, 4, 8], [ 4, 2, 4, 2, 4, 8, 6, 4, 8], [ 4, 6, 2, 6, 4, 2, 4, 6, 8], [ 2, 4, 6, 2, 6, 4, 6, 2, 10], [ 2, 6, 4, 2, 4, 6, 6, 2, 10], [ 2, 6, 4, 2, 4, 2, 10, 2, 10], [ 2, 4, 2, 4, 6, 2, 10, 2, 10], [ 2, 4, 6, 2, 6, 4, 2, 6, 10], [ 2, 4, 2, 4, 6, 2, 6, 6, 10], [ 4, 6, 2, 6, 4, 2, 4, 2, 12], [ 2, 4, 6, 2, 6, 4, 2, 4, 12], [ 2, 4, 2, 4, 6, 2, 6, 4, 12], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 8] 97: [ 2], [ 4], [ 6], [ 8], [ 2, 10], [ 2, 12], [ 2, 6, 10], [ 2, 4, 12], [ 2, 6, 6, 10], [ 2, 4, 8, 10], [ 2, 6, 4, 12], [ 4, 2, 6, 12], [ 2, 4, 6, 12], [ 4, 2, 4, 14], [ 2, 4, 2, 4, 8, 6, 10], [ 2, 4, 2, 4, 6, 8, 10], [ 4, 2, 4, 6, 2, 6, 12], [ 2, 4, 2, 4, 6, 2, 16], [ 2, 4, 2, 4, 6, 2, 6, 6, 4, 12], [ 2, 6, 4, 2, 4, 6, 6, 2, 6, 12], [ 2, 4, 2, 4, 6, 2, 6, 4, 6, 12], [ 2, 4, 2, 4, 6, 2, 10, 2, 4, 14], [ 2, 4, 2, 4, 6, 2, 6, 6, 4, 14], [ 2, 4, 6, 2, 6, 4, 2, 4, 6, 14], [ 2, 4, 2, 4, 6, 2, 6, 4, 6, 14], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 16], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 18], [ 2, 4, 6, 2, 6, 4, 2, 4, 6, 8, 10], [ 2, 4, 6, 2, 6, 4, 2, 4, 2, 12, 10], [ 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 12], [ 2, 4, 6, 2, 6, 4, 2, 4, 2, 10, 12] [/CODE] Computation time: less than a minute, but I believe it is possible to do it in less than a second with some really optimised code. |
Very good! It seems like the popularity of a prime as the start of a maximal gap is based upon the gaps before p. However, there are weird exceptions like 29 and 37. The 2 gaps before 29 are (4, 6) and the gaps before 37 are (2, 6), but 29 is more popular than 37. If (2, 8) was an admissible gap combination, then that would be a forbidden gap combination for 29, but not 37. However, (2, 8) is not admissible.
By the way, I meant to use <= instead of >= in my previous post. The forbidden gap combinations are minimal admissible k-tuples <= the k gaps before p[SUB]m[/SUB]. How do I make the correct symbols for <= and >=? |
1 Attachment(s)
I have finally fully figured out how to tackle the behaviour of [$]o_n(x)[/$] - i.e. the number of occurrences of primes [$]p_n[/$] as initial members of maximal gaps between non-consecutive primes [$]p_n[/$] and [$]p_{n+k}[/$] for all [$]k<=x[/$].
Again, sincere thanks to Bobby for pushing me in the right direction. Although, "forbidden gap constellations" sounds kind of illegal, anyone mind if I call them "blocking patterns" or similar? Suggestions are welcome. So, let [$]B(p_n)[/$] be the set of blocking patterns for the prime [$]p_n[/$], for example [$]B(31)=\lbrace\lbrace0,2\rbrace,\lbrace0,2,6\rbrace,\lbrace0,2,8\rbrace,\lbrace0,2,8,12\rbrace,\lbrace0,2,6,12\rbrace\rbrace[/$]. (Correspondingly, the blocking gap patterns are [$]\lbrace\lbrace2\rbrace,\lbrace2,4\rbrace,\lbrace2,6\rbrace,\lbrace2,6,4\rbrace,\lbrace2,4,6\rbrace\rbrace[/$].) These patterns form a minimal set of sorts. I got temporarily addicted to try and find as many of them as possible. Much to my surprise, I recently even managed to get up to p=97 in less than a second even though my code is far from being optimised, but computation time is ballooning exponentially for larger p. The list in the attachment is not guaranteed to be exhaustive. To evaluate [$]o_n(x)[/$] directly, we subtract from x the number of occurrences of all patterns in [$]B(p_n)[/$] in the range [[$]p_{n-k+2}[/$], [$]p_{n+x}[/$]], where k is the cardinality of the pattern. By looking at the table of blocking patterns, we can now see right away, for example, that 29 occurs more often than 37 for large x by a margin equivalent to the number of occurrences of the patterns {0,2,6,14} and {0,2,6,8,18} below x. This answers post # 44. [$]o_n(x)[/$] remains large if there are very few blocking patterns. [$]n=2[/$] has none because [$]B(p_n)=B(3)=\lbrace0,1\rbrace[/$], a non-admissible prime pattern for [$]p>=3[/$], hence [$]o_1(x)=o_2(x)=x[/$]. [$]n=3[/$] and [$]n=4[/$] only have [$]\lbrace0,2\rbrace[/$] as blocking patterns, all larger n have at least one pattern more, thus [$]o_4(x)>o_n(x)[/$] for all [$]n>4[/$] and [$]x>18[/$] (particularly, [$]o_4(x)=x+2-\#(twin\:primes\:below\:p_{4+x})[/$]). There's [$]n=8[/$] with [$]B(19)=\lbrace\lbrace0,2\rbrace,\lbrace0,2,6\rbrace\rbrace[/$], a minimum for its kind, only twins and the first kind of triplets are blocked, and all [$]n>8[/$] have at least one more blocking pattern, or one that is more common, like [$]\lbrace0,6\rbrace[/$]. We have [$]o_8(x)>o_n(x)[/$] for all [$]n>8[/$] and [$]x>496[/$] (i.e. p=19 "cannot be beaten" above that point). The asymptotic growth rate of [$]o_n(x)[/$] can be obtained via the blocking patterns with additional regard to the open residue classes in each pattern. For x large, [$]o_{29}(x)[/$] ([$]p_{29}=109[/$]) differs from [$]o_8(x)[/$] only by a margin of the number of occurrences of sextuples [$]\lbrace0,2,8,12,14,20\rbrace[/$], [$]\lbrace0,2,6,12,14,20\rbrace[/$], [$]\lbrace0,2,6,8,12,18\rbrace[/$], and [$]\lbrace0,2,6,8,12,20\rbrace[/$] with a total of 8 open residue classes mod 210; in terms of error this is [$]O(x\cdot(\log x)^{-6})[/$]. We can leave septuples or longer patterns out of the game as the have [$]O(x\cdot(\log x)^{-7})[/$] or smaller. If p=109 should be beaten in the long run, it requires, apart from the minimum of [$]\lbrace0,2\rbrace[/$] and [$]\lbrace0,2,6\rbrace[/$] as blocking patterns, either sextuples with less open residue classes in total, or no sextuples at all. And of course, no quadruple or quintuple blocking pattern as well. The next candidate for this is p=5659: only [$]\lbrace0,2,6,8,12,18\rbrace[/$] gets blocked, and this pattern has only one open residue class mod 210. Regarding p=5659 vs. p=9439 (my fallacy in post # 40), the latter seems to be in the lead judging by the small numbers because of the millions of possible blocking patterns in favor of p=9439, but these have a cardinality of as small as 5. At [$]x=10^6[/$], p=9439 is in the lead by more than 30,000 - it takes at least as many quintuples of the forms [$]\lbrace0,4,6,12,16\rbrace[/$], [$]\lbrace0,4,6,10,16\rbrace[/$], [$]\lbrace0,6,8,14,18\rbrace[/$], [$]\lbrace0,2,8,14,18\rbrace[/$], [$]\lbrace0,6,10,12,18\rbrace[/$], [$]\lbrace0,4,10,12,18\rbrace[/$], [$]\lbrace0,6,8,12,18\rbrace[/$], [$]\lbrace0,2,8,12,18\rbrace[/$], [$]\lbrace0,2,6,12,18\rbrace[/$], [$]\lbrace0,4,6,10,18\rbrace[/$], or [$]\lbrace0,2,6,8,18\rbrace[/$] until p=5659 can overtake p=9439, we expect this not to happen before [$]x=10^8[/$]. To conclude, the primes for which a local maximum as described in post # 35 is reached for [$]\lim x\to\infty[/$], or rather, for sufficiently large x, should be equal to [$]2\:(3), 7,[/$] and [$]19[/$], with infinitely many [$]o_n(x)[/$] for [$]n>8[/$] coming arbitrarily close to [$]o_8(x)-O(x\cdot(\log x)^{-6})[/$] (e.g. the primes [$]5659[/$] ([$]n=746[/$]), [$]88819[/$] ([$]n=8605[/$]), [$]855739[/$] ([$]n=68032[/$]), [$]74266279[/$] ([$]n=4353833[/$]), [$]964669639[/$] ([$]n=49141276[/$]), [$]9853497769[/$] ([$]n=448687813[/$]), etc. each move toward this upper bound from below). So my previous implicit assumption that the list of primes with local maxima, bounded from above, is infinite was wrong. Phew, that took long enough. And it's only framework, sort of. Also poorly worded at times, but I really need to finish this off now, one way or another. I flip out if now someone gives me a link to some obscure 19th century work that covers all this... |
"Sitting target / sitting, waiting / anticipating / nothing / nothing."
CSG looks well-behaved even for k <= 1024 (p in range < 10^14):
[CODE] k gap CSG_max p 112 5940 1.05550107 36683716323847 116 6052 1.02516052 36683716323619 120 6220 1.03269957 36683716323283 124 6388 1.04043735 36683716323283 128 6510 1.01858817 36683716323161 132 6642 1.00390061 36683716323167 136 6742 0.96976743 36683716323109 140 6658 0.94384648 17674627574311 144 6840 0.96488178 17674627574369 148 6992 0.96688912 17674627574141 152 7126 0.95779790 17674627574083 156 7460 0.97452838 30512335335437 160 7614 0.97643675 30512335335437 164 7732 0.95708911 30512335335319 168 7946 0.99499058 30512335334951 172 8100 0.99726110 30512335334797 176 8254 0.99967015 30512335334797 180 8364 0.97661645 30512335335299 184 8510 0.97483498 30512335335059 188 8736 1.01921576 30512335334927 192 8892 1.02319197 30512335334771 196 9004 1.00215904 30512335334797 200 9148 1.01448166 28330683392731 204 9324 1.03039259 28330683392659 208 9492 1.04177291 28330683392597 212 9630 1.03626675 28330683392353 216 9778 1.03654152 28330683392371 220 9856 0.99828866 28330683392371 224 9974 0.98269325 28330683392147 228 10058 0.94929275 28330683392129 232 8294 0.94835143 185067241757 236 9700 0.95641246 5185992136441 240 9850 0.96394780 5185992136453 244 10626 0.94205155 28330683392597 248 10818 0.96644566 28330683392371 252 10596 0.94771341 12666866223047 256 11310 0.93908073 52248744686339 260 11476 0.94818065 52248744686197 264 11604 0.93866201 52248744686069 268 11724 0.92547977 52248744686197 272 11264 0.93001522 12666866223047 276 12106 0.91167574 68182243872601 280 11752 0.91251084 21947823205027 284 11920 0.92535164 21947823205027 288 12096 0.94216306 21947823204943 292 12310 0.94178965 25698372297889 296 12460 0.94533825 25698372297691 300 12704 0.99505355 25698372297029 304 12920 1.03124573 25698372297029 308 13170 1.08475215 25698372297029 312 13308 1.08194976 25698372297029 316 13482 1.09729159 25698372297029 320 13616 1.09257083 25698372296963 324 13728 1.07704353 25698372296873 328 13878 1.08048911 25698372297029 332 13986 1.06336781 25698372297007 336 14136 1.06693878 25698372296963 340 14234 1.04538481 25698372296873 344 14336 1.02612587 25698372296243 348 14466 1.02043217 25698372295733 352 14642 1.03657564 25698372297029 356 14778 1.03379163 25698372295733 360 14890 1.01983645 25698372295711 364 15044 1.02563857 25698372296963 368 15222 1.04265118 25698372295019 372 15360 1.04099593 25698372294839 376 15546 1.06172759 25698372295033 380 15694 1.06474083 25698372294457 384 15832 1.06313720 25698372294409 388 15968 1.06066576 25698372294611 392 16158 1.08316800 25698372294421 396 16242 1.05682221 25698372294337 400 16344 1.03908815 25698372294563 404 16536 1.06228245 25698372294457 408 16678 1.06277691 25698372294421 412 16762 1.03722225 25698372294337 416 16852 1.01477000 25698372294457 420 16974 1.00672310 25698372295033 424 17160 1.02688378 25698372294421 428 17302 1.02766540 25698372294421 432 17396 1.00751345 25698372294611 436 17586 1.02931311 25698372294421 440 17724 1.02843581 25698372294409 444 17810 1.00513477 25698372294323 448 17886 0.97797632 25698372294253 452 17972 0.95549900 25698372294281 456 18114 0.95669262 25698372293557 460 18234 0.94875594 25698372293809 464 18390 0.95581939 25698372293597 468 18536 0.95873888 25698372293597 472 19506 0.94029942 93152147737543 476 19770 0.98628553 93152147737279 480 19878 0.97192308 93152147737199 484 19954 0.94554667 93152147737237 488 19192 0.94334862 25698372294421 492 20322 0.97552201 93152147736727 496 20490 0.98440496 93152147736559 500 20598 0.97040570 93152147736451 504 20748 0.97245252 93152147736301 508 20850 0.95643446 93152147736199 512 21004 0.96004428 93152147736073 516 21260 1.00210684 93152147735789 520 21390 0.99658464 93152147735659 524 21478 0.97538922 93152147735599 528 21592 0.96413821 93152147735371 532 21726 0.96039757 93152147735351 536 21874 0.96185682 93152147735203 540 21964 0.94210589 93152147735113 544 22076 0.93058797 93152147734973 548 22224 0.93216222 93152147733739 552 22486 0.97513465 93152147732647 556 22628 0.97445048 93152147734421 560 22792 0.98180763 93152147734171 564 22958 0.98989507 93152147734091 568 23130 1.00017583 93152147733919 572 23346 1.02661845 93152147733703 576 23524 1.03914014 93152147733553 580 23610 1.01786364 93152147733467 584 23706 1.00050490 93152147733553 588 23912 1.02309904 93152147733137 592 24068 1.02754489 93152147732981 596 24240 1.03781136 93152147732723 598 24402 1.07077256 93152147732647 600 24436 1.05684370 93152147732641 604 24540 1.04234724 93152147732509 608 24676 1.03956218 93152147732401 612 24798 1.03177448 93152147732251 616 24880 1.00980556 93152147732197 620 25008 1.00437652 93152147732069 624 25164 1.00888491 93152147731913 628 25264 0.99368357 93152147731813 632 25348 0.97310855 93152147731729 636 25500 0.97626664 93152147731549 640 25578 0.95395507 93152147731499 644 25696 0.94554893 93152147731381 648 25860 0.95284382 93152147731217 652 26004 0.95333346 93152147731073 656 26252 0.98935642 93152147730797 660 26412 0.99528904 93152147730637 664 26606 1.01294463 93152147730443 668 26706 0.99822651 93152147730371 672 26826 0.99048613 93152147730223 676 26938 0.98010919 93152147730139 680 27094 0.98468729 93152147729983 684 27186 0.96770698 93152147729891 688 27276 0.95027202 93152147729983 692 27368 0.93371537 93152147729891 696 27516 0.93569220 93152147729561 700 27582 0.91092486 93152147729467 704 27698 0.90261024 93152147729561 708 27820 0.89629956 93152147729143 712 27948 0.89196235 93152147729143 716 28048 0.87877867 93152147729143 720 27710 0.89275685 54116590394771 724 27860 0.89636501 54116590394621 728 27998 0.89606095 54116590394483 732 28172 0.90750139 54116590393157 736 28332 0.91437757 54116590394149 740 28536 0.93570243 54116590393991 744 28666 0.93272540 54116590393861 748 28800 0.93108486 54116590393777 752 28982 0.94518150 54116590393499 756 29130 0.94812537 54116590393447 760 29370 0.98143984 54116590393157 764 29456 0.96387023 54116590393121 768 29630 0.97537452 54116590392947 772 29706 0.95469068 54116590393157 776 29826 0.94853391 54116590392947 780 29964 0.94825668 54116590392929 784 30076 0.93959664 54116590392451 788 30192 0.93229670 54116590392947 792 30288 0.91869099 54116590392289 796 30456 0.92805971 54116590392121 800 30654 0.94703768 54116590391873 804 30740 0.93024784 54116590391837 808 30816 0.91051339 54116590391861 812 30990 0.92171618 54116590391873 816 31176 0.93673527 54116590391351 820 31368 0.95371131 54116590391113 824 31516 0.95671400 54116590391011 828 31596 0.93820121 54116590391011 832 31734 0.93807554 54116590391113 836 31852 0.93169439 54116590391011 840 31936 0.91480473 54116590391077 844 32062 0.91104489 54116590391077 848 32158 0.89810577 54116590391011 852 32880 0.89162466 93152147732647 856 33006 0.88736277 93152147732641 860 32594 0.90483328 54116590389887 864 32714 0.89936117 54116590389863 868 32790 0.88065023 54116590389887 872 32960 0.89035364 54116590389887 876 33068 0.88139835 54116590389473 880 33158 0.86716391 54116590389419 884 33276 0.86135519 54116590389863 888 33420 0.86326354 54116590389473 892 33550 0.86104325 54116590388977 896 33738 0.87595633 54116590388789 900 34052 0.86261430 65480290959731 904 34264 0.88413463 65480290959547 908 34380 0.87757423 65480290959403 912 34474 0.86468039 65480290959403 916 35030 0.86491644 93152147730443 920 35156 0.86099754 93152147730497 924 34932 0.87719975 65480290958651 928 35160 0.90333082 65480290958651 932 35254 0.89038566 65480290958557 936 35504 0.89448345 70981263873617 940 35646 0.89544859 70981263873617 944 35654 0.88610601 65480290958129 948 35814 0.89239499 65480290957997 952 36080 0.90067205 70981263873617 956 36204 0.89647169 70981263873617 960 36294 0.88259124 70981263872969 964 32722 0.87857798 3529553758999 968 36550 0.87667128 70981263873109 972 36728 0.88790229 70981263872969 976 36864 0.88721696 70981263872257 980 37006 0.88823910 70981263872257 984 37086 0.87179868 70981263872257 988 37224 0.87172780 70981263872257 992 37440 0.89358331 70981263872257 996 37564 0.88954266 70981263872257 1000 36346 0.87913969 25264345114117 1004 36534 0.89520659 25264345113919 1008 36604 0.87646499 25264345113919 1012 36740 0.87719292 25264345113713 1016 36876 0.87792446 25264345113613 1020 37294 0.87603752 31618998499597 1024 37074 0.85792826 25264345113613 [/CODE] Update on blocking patterns (see previous post): p = 157: 1195 patterns on my watch p = 163: at least 2125 patterns p = 167: at least 4000 patterns p = 173: at least 5733 patterns p = 179: at least 7357 patterns p = 181: at least 16345 patterns p = 191: at least 11710 patterns But the number of patterns is not terribly important (and probably impossible to compute in full for p > 179 or 181) - for some decent comparisons between values of o[SUB]n[/SUB](x), it should be sufficient to know the patterns with cardinality <= 7 or 8 or thereabouts, these are not too hard to figure out if p is not too large. This is getting boring, I'm going to watch some episodes of PJ Masks now.:popcorn: |
[QUOTE=Bobby Jacobs;604770]What is the pattern with the sequence of primes with record low numbers of occurrences? It seems like the sequence is 2, 5, 11, 29, 37, 59, 97, 127, 223, 307, 541, 907, 1151, 1361, ... This is similar to the primes at the end of maximal prime gaps, but not exactly. I wonder what the pattern is.[/QUOTE]
I believe that 223 will eventually beat 127. It starts out behind because there are two consecutive gaps of 12 before 223. However, 223 will eventually catch up with 127 because 127 has a gap of 14. Therefore, the sequence of record lows will start 2, 5, 11, 29, 37, 59, 97, 127, 307, 541, 907, 1151, 1361, ... |
[QUOTE=Bobby Jacobs;607677]I believe that 223 will eventually beat 127. It starts out behind because there are two consecutive gaps of 12 before 223. However, 223 will eventually catch up with 127 because 127 has a gap of 14. Therefore, the sequence of record lows will start 2, 5, 11, 29, 37, 59, 97, 127, 307, 541, 907, 1151, 1361, ...[/QUOTE]
That's what I would assume as well. These would be the primes at the end of a maximal gap, including ones where there is a tie to the previous maximal gap, if the blocking patterns cover more common patterns. My search is still running, slowly approaching 2e14 for k <= 109. Does anyone think CSG > 1.1 is possible to find? |
Don't answer me, don't break the silence, don't let me win
1 Attachment(s)
Attached are the numbers of first occurrence gaps for k <= 109 and p <= 2e14.
Don't look for me, I'm already moving on.:digging: |
Error terms, prime number scarcities, and trains
Just an intermediate result that made me go "hmmmm...". Suppose we assume [$]CSG=1+O(1)[/$] for the gaps between non-consecutive primes, then, if I did the math right, this would imply that we also assume [$]\pi(x)=Li(x)+O(\sqrt{x})[/$], i.e. the error term is smaller by a factor log x compared to the RH prediction. Correct [y/n]?
[CODE]Outline from my train of thought: p_1 = 2 (or set p_0 = 0, say) p_k = x k = pi(x)-1 ~ pi(x) gap = x-2 ~ x m = Gram(x)-Gram(2)-k+1 ~ Gram(x)-pi(x) CSG = m*|m|/gap - but for simplicity suppose that m is positive (means we assume a scarcity instead of an abundance of primes; the error term works both ways anyway): CSG = m^2/gap ~ (Gram(x)-pi(x))^2/x CSG ~ 1 --> (Gram(x)-pi(x))^2 ~ x --> Gram(x)-pi(x) ~ sqrt(x) OTOH, if Gram(x)-pi(x) = O(sqrt(x)*log(x)), then CSG = m^2/gap ~ O((x*log²x)/x) ~ O(log²x)[/CODE] |
That is correct. By the way, you should submit the sequences in this thread to OEIS.
|
1 Attachment(s)
[QUOTE=mart_r;596734]
As a by-product, a puzzle: Given x, find the next three consecutive primes >= x. Denote the two gaps between them g[SUB]1[/SUB] and g[SUB]2[/SUB], and let g[SUB]1[/SUB] >= g[SUB]2[/SUB]. Let r = g[SUB]1[/SUB]/g[SUB]2[/SUB]. As x becomes larger, the geometric mean r[SUB]gm[/SUB] of values of r also become larger. Find an asymptotic function f(x) ~ r[SUB]gm[/SUB].[/QUOTE] [$]\lim\limits_{\substack{x\to\infty}} r_{gm} = 4[/$], if I may so conjecture (based on a random model similar to Cramér's). Is there a proof available? I'd like to take the search for T(38,16) in A[OEIS]086153[/OEIS] up to 10^16, which will likely not be enough to find an example, but I still would like to see that case solved. It would take a bit more than a week with my program. I've identified 746 distinct constellations as shown in the attachment. I believe that list to be complete, albeit I'd be more content if that number was divisible by 4, so there's a slight possibility I have overlooked some constellations. If anyone with enough time on their hands feels inclined to do a quick double-check... For good measure, here's a batch of 79 instances where CSG > 1 for k > 1000: [CODE]p k gap CSG 123146152018999 1152 44280 1.0322718 123146152018933 1154 44346 1.0310658 123146152018999 1127 43378 1.0263032 123146152018999 1126 43342 1.0260889 123146152018933 1129 43444 1.0250767 123146152018933 1156 44394 1.0249425 123146152018933 1128 43408 1.0248616 123146152018921 1155 44358 1.0247205 123146152018999 1157 44428 1.0246186 123146152018999 1133 43582 1.0242866 123146152018993 1153 44286 1.0242773 123146152018999 1138 43758 1.0242719 123146152018933 1159 44494 1.0234272 123146152018933 1135 43648 1.0230694 123146152018933 1140 43824 1.0230603 123146152018823 1158 44456 1.0226588 123146152019071 1151 44208 1.0221944 123146152018999 1125 43288 1.0209057 123146152018999 1139 43780 1.0206462 123146152018933 1141 43846 1.0194401 123146152018933 1160 44514 1.0192932 123146152018933 1130 43458 1.0192328 123146152019521 1113 42856 1.0183353 123146152019521 1112 42820 1.0181224 123146152018853 1132 43524 1.0180185 123146152018853 1131 43488 1.0178006 123146152018999 1094 42184 1.0176670 123146152018999 1091 42078 1.0176016 123146152018921 1136 43660 1.0166975 123146152019521 1143 43906 1.0165963 123146152018853 1162 44574 1.0164815 123146152018933 1096 42250 1.0164132 123146152018933 1093 42144 1.0163444 123146152019521 1119 43060 1.0163041 123146152019521 1124 43236 1.0162807 123146152018993 1134 43588 1.0162592 123146152018999 1095 42210 1.0150911 123146152018823 1163 44604 1.0150789 123146152018823 1144 43934 1.0146314 123146152019071 1137 43686 1.0141775 123146152018933 1097 42276 1.0138418 123146152019419 1142 43860 1.0136414 123146152018823 1161 44528 1.0135410 123146152019521 1111 42766 1.0129308 123146152019507 1114 42870 1.0124697 123146152018801 1164 44626 1.0115109 123146152019521 1116 42936 1.0112458 123146152018801 1145 43956 1.0110334 123146152019507 1120 43074 1.0104594 123146152019521 1080 41662 1.0097438 123146152019521 1077 41556 1.0096842 123146152018993 1092 42084 1.0094570 123146152019483 1115 42894 1.0093765 123146152018853 1099 42330 1.0092682 123146152018999 1088 41938 1.0080624 123146152018823 1100 42360 1.0078183 123146152019419 1117 42958 1.0076073 123146152018801 1146 43978 1.0074438 123146152018801 1165 44646 1.0074105 123146152018921 1098 42288 1.0073935 123146152019483 1121 43098 1.0073785 123146152019071 1090 42006 1.0073695 123146152019521 1081 41688 1.0071605 123146152019521 1079 41616 1.0067457 123146152019419 1147 44008 1.0060364 123146152019167 1150 44112 1.0056258 123146152019419 1123 43162 1.0056203 123146152018823 1101 42386 1.0052645 123146152019207 1149 44072 1.0043139 123146152018999 1089 41958 1.0038262 123146152019461 1122 43120 1.0037558 123146152019507 1078 41570 1.0037494 123146152019521 1118 42976 1.0028794 123146152018583 1167 44696 1.0019334 123146152019419 1148 44028 1.0019166 123146152018801 1102 42408 1.0016116 123146152019507 1082 41702 1.0012434 123146152019521 1074 41416 1.0001307 123146152018793 1166 44654 1.0000862 [/CODE] |
Dancing with tears in my eyes...
[QUOTE=mart_r;612010][$]\lim\limits_{\substack{x\to\infty}} r_{gm} = 4[/$], if I may so conjecture (based on a random model similar to Cramér's). Is there a proof available?
[/QUOTE] Is this not a well-known result? If so, I might try to tackle the proof myself... They say that the gas pipeline Nord Stream 1 is kept shut indefinitely. So before power outrages become daily routine, I'd like to give an update on some numbers. - Gaps between non-consecutive primes, for k=104 to 1024 step 4: [CODE]p <= 179133400000000 k gap CSG_max p 104 5656 1.0781126752 36683716323847 108 5824 1.0843154811 36683716323847 112 5940 1.0555010733 36683716323847 116 6052 1.0251605182 36683716323619 120 6220 1.0326995729 36683716323283 124 6388 1.0404373460 36683716323283 128 6510 1.0185881717 36683716323161 132 6642 1.0039006107 36683716323167 136 6742 0.9697674279 36683716323109 140 7292 0.9521920888 175478559288359 144 6840 0.9648817776 17674627574369 148 6992 0.9668891162 17674627574141 152 7126 0.9577979013 17674627574083 156 7460 0.9745283792 30512335335437 160 8144 0.9792679542 175478559288359 164 7732 0.9570891069 30512335335319 168 7946 0.9949905766 30512335334951 172 8100 0.9972610993 30512335334797 176 8254 0.9996701497 30512335334797 180 8364 0.9766164520 30512335335299 184 8510 0.9748349809 30512335335059 188 8736 1.0192157620 30512335334927 192 8892 1.0231919672 30512335334771 196 9004 1.0021590389 30512335334797 200 9148 1.0144816568 28330683392731 204 9324 1.0303925866 28330683392659 208 9492 1.0417729138 28330683392597 212 9630 1.0362667509 28330683392353 216 9778 1.0365415158 28330683392371 220 9856 0.9982886575 28330683392371 224 9974 0.9826932475 28330683392147 228 10058 0.9492927463 28330683392129 232 8294 0.9483514306 185067241757 236 9700 0.9564124562 5185992136441 240 9850 0.9639477964 5185992136453 244 10626 0.9420515461 28330683392597 248 10818 0.9664456553 28330683392371 252 10596 0.9477134052 12666866223047 256 11310 0.9390807333 52248744686339 260 11476 0.9481806463 52248744686197 264 11604 0.9386620116 52248744686069 268 11724 0.9254797708 52248744686197 272 11264 0.9300152206 12666866223047 276 12106 0.9116757369 68182243872601 280 11752 0.9125108382 21947823205027 284 11920 0.9253516441 21947823205027 288 12096 0.9421630582 21947823204943 292 12310 0.9417896525 25698372297889 296 12460 0.9453382500 25698372297691 300 12704 0.9950535482 25698372297029 304 12920 1.0312457318 25698372297029 308 13170 1.0847521505 25698372297029 312 13308 1.0819497629 25698372297029 316 13482 1.0972915901 25698372297029 320 13616 1.0925708301 25698372296963 324 13728 1.0770435304 25698372296873 328 13878 1.0804891085 25698372297029 332 13986 1.0633678090 25698372297007 336 14136 1.0669387810 25698372296963 340 14234 1.0453848066 25698372296873 344 15204 1.0287670437 127946496635897 348 15390 1.0445817969 127946496635761 352 15540 1.0446056952 127946496635611 356 15692 1.0455495832 127946496635459 360 15798 1.0266221073 127946496635459 364 15044 1.0256385680 25698372296963 368 15222 1.0426511752 25698372295019 372 15360 1.0409959251 25698372294839 376 15546 1.0617275885 25698372295033 380 15694 1.0647408322 25698372294457 384 15832 1.0631372016 25698372294409 388 15968 1.0606657649 25698372294611 392 16158 1.0831680032 25698372294421 396 16242 1.0568222056 25698372294337 400 16344 1.0390881483 25698372294563 404 16536 1.0622824463 25698372294457 408 16678 1.0627769150 25698372294421 412 16762 1.0372222537 25698372294337 416 16852 1.0147699971 25698372294457 420 16974 1.0067230990 25698372295033 424 17160 1.0268837758 25698372294421 428 17302 1.0276653950 25698372294421 432 17396 1.0075134540 25698372294611 436 17586 1.0293131103 25698372294421 440 17724 1.0284358125 25698372294409 444 17810 1.0051347652 25698372294323 448 17886 0.9779763169 25698372294253 452 17972 0.9554989984 25698372294281 456 18114 0.9566926179 25698372293557 460 18234 0.9487559428 25698372293809 464 18390 0.9558193895 25698372293597 468 18536 0.9587388800 25698372293597 472 19656 0.9486426201 112364701413971 476 19770 0.9862855261 93152147737279 480 19878 0.9719230760 93152147737199 484 19954 0.9455466698 93152147737237 488 19192 0.9433486203 25698372294421 492 20322 0.9755220092 93152147736727 496 20490 0.9844049639 93152147736559 500 20598 0.9704056982 93152147736451 504 20748 0.9724525235 93152147736301 508 20850 0.9564344564 93152147736199 512 21004 0.9600442813 93152147736073 516 21260 1.0021068423 93152147735789 520 21390 0.9965846355 93152147735659 524 21478 0.9753892159 93152147735599 528 21592 0.9641382100 93152147735371 532 21726 0.9603975747 93152147735351 536 21874 0.9618568153 93152147735203 540 21964 0.9421058879 93152147735113 544 22076 0.9305879709 93152147734973 548 22224 0.9321622171 93152147733739 552 22486 0.9751346521 93152147732647 556 22628 0.9744504782 93152147734421 560 22792 0.9818076325 93152147734171 564 22958 0.9898950736 93152147734091 568 23130 1.0001758329 93152147733919 572 23346 1.0266184470 93152147733703 576 23524 1.0391401441 93152147733553 580 23610 1.0178636352 93152147733467 584 23706 1.0005048976 93152147733553 588 23912 1.0230990399 93152147733137 592 24068 1.0275448938 93152147732981 596 24240 1.0378113630 93152147732723 600 24436 1.0568436976 93152147732641 604 24540 1.0423472371 93152147732509 608 24676 1.0395621760 93152147732401 612 24798 1.0317744784 93152147732251 616 24880 1.0098055650 93152147732197 620 25008 1.0043765193 93152147732069 624 25164 1.0088849117 93152147731913 628 25264 0.9936835657 93152147731813 632 25348 0.9731085473 93152147731729 636 25500 0.9762666351 93152147731549 640 25578 0.9539550727 93152147731499 644 25696 0.9455489291 93152147731381 648 25860 0.9528438164 93152147731217 652 26004 0.9533334554 93152147731073 656 26252 0.9893564159 93152147730797 660 26412 0.9952890412 93152147730637 664 26606 1.0129446324 93152147730443 668 26706 0.9982265121 93152147730371 672 26826 0.9904861287 93152147730223 676 26938 0.9801091903 93152147730139 680 27094 0.9846872883 93152147729983 684 27186 0.9677069785 93152147729891 688 27276 0.9502720219 93152147729983 692 27368 0.9337153701 93152147729891 696 27516 0.9356922025 93152147729561 700 27582 0.9109248623 93152147729467 704 27698 0.9026102400 93152147729561 708 27820 0.8962995612 93152147729143 712 27948 0.8919623539 93152147729143 716 28048 0.8787786737 93152147729143 720 27710 0.8927568473 54116590394771 724 27860 0.8963650091 54116590394621 728 27998 0.8960609520 54116590394483 732 28172 0.9075013936 54116590393157 736 28332 0.9143775739 54116590394149 740 28536 0.9357024342 54116590393991 744 28666 0.9327253951 54116590393861 748 28800 0.9310848576 54116590393777 752 28982 0.9451815024 54116590393499 756 29130 0.9481253683 54116590393447 760 29370 0.9814398420 54116590393157 764 29456 0.9638702316 54116590393121 768 29630 0.9753745155 54116590392947 772 29706 0.9546906758 54116590393157 776 29826 0.9485339103 54116590392947 780 29964 0.9482566835 54116590392929 784 30076 0.9395966369 54116590392451 788 30192 0.9322967009 54116590392947 792 30288 0.9186909882 54116590392289 796 30456 0.9280597127 54116590392121 800 30654 0.9470376796 54116590391873 804 30740 0.9302478398 54116590391837 808 31974 0.9117548600 159316577936029 812 30990 0.9217161811 54116590391873 816 31176 0.9367352659 54116590391351 820 32550 0.9554814455 159316577935453 824 31516 0.9567140006 54116590391011 828 31596 0.9382012070 54116590391011 832 31734 0.9380755356 54116590391113 836 31852 0.9316943869 54116590391011 840 31936 0.9148047257 54116590391077 844 32062 0.9110448935 54116590391077 848 32158 0.8981057702 54116590391011 852 32880 0.8916246643 93152147732647 856 33006 0.8873627713 93152147732641 860 32594 0.9048332804 54116590389887 864 32714 0.8993611673 54116590389863 868 34120 0.9116132357 159316577935453 872 34218 0.8986902090 159316577935453 876 34398 0.9093907279 159316577935453 880 34498 0.8971206895 159316577935453 884 34592 0.8832711645 159316577933411 888 34758 0.8899100180 159316577935453 892 34934 0.8993959518 159316577936297 896 35074 0.8986493801 159316577936233 900 35262 0.9115556093 159316577935969 904 35406 0.9119383256 159316577935453 908 35630 0.9351767250 159316577935601 912 35784 0.9384016321 159316577935453 916 35880 0.9250502187 159316577935453 920 36024 0.9254373323 159316577935433 924 36102 0.9071761352 159316577935601 928 36288 0.9194386246 159316577935453 932 36460 0.9277658577 159316577935453 936 36576 0.9202752667 159316577935601 940 36744 0.9274602593 159316577935453 944 36906 0.9329559987 159316577935453 948 37068 0.9384527900 159316577935453 952 37140 0.9186590345 159316577935453 956 37282 0.9185545605 159316577935969 960 37384 0.9073477992 159316577935969 964 37650 0.9418370777 159316577935601 968 37800 0.9439561470 159316577935451 972 37914 0.9360034504 159316577935423 976 38014 0.9242274696 159316577935969 980 37812 0.9276107833 120293264372867 984 38382 0.9474156903 159316577935601 988 38538 0.9512197726 159316577935453 992 38680 0.9511189880 159316577935453 996 38796 0.9438031196 159316577935453 1000 38866 0.9238826098 159316577935453 1004 39040 0.9326313213 159316577935453 1008 39126 0.9172648307 159316577935453 1012 39348 0.9391390323 159316577935601 1016 39538 0.9523097203 159316577935453 1020 39680 0.9522520689 159316577935601 1024 39856 0.9615749322 159316577935453 [/CODE] - Gaps between primes in arithmetic progression, for q=4568 to 5004 step 2: [CODE]p <= 23388300000000 q gap CSG (conv.) p 4568 1548552 0.8572222356 1677084447851 4570 1183630 0.8390238160 1196563621633 4572 676656 0.9313083686 3312086153 4574 864486 0.8348603294 1749438037 4576 617760 0.8078001685 464384941 4578 691278 0.8429669902 84048460189 4580 1108360 0.8026462441 890252180611 4582 1383764 0.8499365180 720395477939 4584 1141416 0.7964331851 21652890442697 4586 1371214 0.8127600152 606453831427 4588 1601212 0.8876812908 3550398242161 4590 867510 0.8724168546 5747636061659 4592 1428112 0.8464431695 7482931558789 4594 1745720 0.8491640934 9894775021751 4596 896220 0.8192055813 417572047247 4598 1549526 0.8296428911 21799960507001 4600 1283400 0.7977381722 13502858057147 4602 1090674 0.8445938334 16897337246939 4604 1749520 0.8362905821 12528155746207 4606 428358 0.8045087092 15779549 4608 926208 0.8881770694 207016317479 4610 1212430 0.8526133843 1183984521847 4612 894728 0.9475156489 618769103 4614 461400 0.8323251172 177577073 4616 1583288 0.8432163277 2500655788181 4618 1454670 0.8260526488 991355520937 4620 328020 0.8965801195 300426827 4622 1687030 0.8202913337 9089241698347 4624 550256 0.8642679215 26284901 4626 1050102 0.8210523101 3403774701511 4628 1596660 0.8144418081 17046531702059 4630 1361220 0.7957535391 16338258362707 4632 291816 0.8541090988 2708653 4634 1181670 0.8315601474 431161340839 4636 1070916 0.7908290917 74835258193 4638 834840 0.8481347282 92370239231 4640 389760 0.8488446783 8560609 4642 1425094 0.8402156795 2200086910369 4644 1114560 0.8332081592 8272678130681 4646 1681852 0.8229584664 17244249871939 4648 957488 0.8394410220 28539747311 4650 916050 0.8669161631 7715831183377 4652 907140 0.8022040760 3800111563 4654 837720 0.9111719565 1022854519 4656 1164000 0.8197000840 16031665116419 4658 1481244 0.7793115567 6846908799389 4660 1337420 0.8169553941 7910942491217 4662 731934 0.7990676643 351440070953 4664 1455168 0.7900947961 8378888283431 4666 1843070 0.8398134556 21034084207667 4668 606840 0.7997013450 4009021879 4670 448320 0.8325428315 23630069 4672 1191360 0.8690884709 39202962193 4674 1051650 0.8288516679 7787632946129 4676 827652 0.8363384979 4784767627 4678 1468892 0.8470457965 672647311367 4680 336960 0.8186005024 161618917 4682 435426 0.8382468056 2520173 4684 1311520 0.7852421447 400661984803 4686 1077780 0.8318802396 16277427487337 4688 1392336 0.8603480773 269749617047 4690 1069320 0.8882226297 939523507907 4692 1102620 0.8986838410 6608920918927 4694 1825966 0.9167594012 4511534178661 4696 723184 0.8224917628 257098579 4698 361746 0.8481017107 19332767 4700 1207900 0.8281666227 1687877098657 4702 893380 0.8660946502 1254661021 4704 940800 0.7810442832 10036010128121 4706 1369446 0.8135118655 1330625650261 4708 630872 0.8454641507 139903367 4710 527520 0.8361774196 5812980889 4712 1526688 0.8205824387 5570936685233 4714 900374 0.8147104625 2546122913 4716 429156 0.7783377852 145721231 4718 1387092 0.8456133333 2444163196961 4720 1368800 0.8525735781 5931853116047 4722 798018 0.8208499645 63126509689 4724 1927392 0.9137557146 9631507165417 4726 1436704 0.8474484525 1081554964909 4728 997608 0.8826331534 457159963609 4730 1177770 0.7925444628 8252709146287 4732 723996 0.8275560049 2446077173 4734 1013076 0.7972189748 2227503505511 4736 374144 0.8089251738 1049177 4738 1331378 0.8449824963 322102867019 4740 616200 0.8374867911 35079078883 4742 1019530 0.8251804802 8239994179 4744 1892856 0.8523019576 19958061839741 4746 322728 0.8242463704 25539551 4748 356100 0.8672257307 161611 4750 1268250 0.8645165155 2502352535699 4752 1092960 0.8514531711 9258938335169 4754 1231286 0.7892730195 134353071551 4756 1112904 0.8082974840 58507371397 4758 494832 0.8244859283 734454311 4760 985320 0.7894969010 2395919782511 4762 1219072 0.9308036050 15409579657 4764 452580 0.8622914404 79985923 4766 1572780 0.8311913693 1739547200591 4768 1587744 0.8131646687 2956675908829 4770 887220 0.7789200810 13193937699571 4772 691940 0.8098206509 165985607 4774 1260336 0.8112539172 5739683343047 4776 448944 0.8442629551 90180059 4778 501690 0.9497720783 2376683 4780 1104180 0.8132485059 395693430791 4782 526020 0.8161806515 546710977 4784 1368224 0.8174372120 1683148638257 4786 789690 0.8598555091 322654259 4788 995904 0.8147188358 21770871850033 4790 1216660 0.8416658242 873768458473 4792 1269880 0.8330024589 92005688941 4794 661572 0.7960364440 20861219971 4796 1251756 0.8307730973 295346729533 4798 1679300 0.8446785055 3197716234463 4800 969600 0.8209328289 15571822465201 4802 388962 0.8869542712 1796987 4804 1710224 0.7782874527 13842179098073 4806 1206306 0.8333516049 13448938011913 4808 533688 0.8537185219 9679121 4810 274170 0.8519340567 570697 4812 307968 0.8415538293 3390703 4814 1473084 0.8271424430 1245798692843 4816 1247344 0.7877295603 1484176131853 4818 876876 0.8215487463 666432600907 4820 935080 0.8313203473 32487008371 4822 1750386 0.8577540614 4340143171271 4824 1201176 0.8698188410 6655443895757 4826 482600 0.8212477031 9305063 4828 1670488 0.7931469867 20747292365671 4830 454020 0.8785153116 4050952519 4832 1797504 0.8485095387 7995365520743 4834 1672564 0.8896834577 1301997325459 4836 614172 0.8180428160 8250971321 4838 1465914 0.7713956943 2688844111963 4840 1316480 0.8550815866 6997095903977 4842 1074924 0.8840456566 875855956663 4844 465024 0.8806644242 8374607 4846 688132 0.8601739780 77467669 4848 1110192 0.7856487086 8063436375157 4850 388000 0.8444615696 4839613 4852 1572048 0.8412827663 1143215029679 4854 961092 0.7804868212 973772706631 4856 1592768 0.8380868777 1446945776483 4858 1462258 0.8207158595 5283274725721 4860 704700 0.7812851332 286560716669 4862 1604460 0.8991146486 17381205974537 4864 812288 0.8810163626 486408491 4866 1124046 0.8356057290 3270645941561 4868 1630780 0.8590928041 1359358668541 4870 1344120 0.8735751658 1652489415169 4872 954912 0.8590073654 3091414128029 4874 731100 0.9439063939 54741157 4876 1482304 0.8250557169 1478412698227 4878 1034136 0.8224074757 1257853715861 4880 683200 0.7939954643 1562016139 4882 1635470 0.8558980255 1423810702421 4884 434676 0.8333452034 183889271 4886 747558 0.8734681702 619568773 4888 821184 0.7856745434 2810773463 4890 322740 0.9302432611 12434311 4892 826748 0.8516229144 451642253 4894 1747158 0.8243990452 6075857766151 4896 563040 0.8478787257 1071146539 4898 1077560 0.8095032996 22817244383 4900 764400 0.8228894964 16299646609 4902 676476 0.7730046246 28069830563 4904 818968 0.8186527675 600834991 4906 1501236 0.8367576017 2219095818727 4908 1173012 0.7979230970 10827220170911 4910 574470 0.8515239444 113538983 4912 1792880 0.8327591122 7574725685297 4914 545454 0.8596639821 4067682527 4916 1042192 0.8689515451 3954781777 4918 1091796 0.7998596761 17149465633 4920 319800 0.7628511092 72051739 4922 1717778 0.7963065272 16173410263259 4924 797688 0.7994122308 557381641 4926 1054164 0.8376718240 1072468422437 4928 1232000 0.8213657477 1376131327733 4930 1296590 0.8304159408 6598307178041 4932 1203408 0.7907077203 18298578679613 4934 1845316 0.7993224339 19412314489657 4936 1584456 0.8193829451 1466754802973 4938 1338198 0.8687111129 19679180418991 4940 350740 0.8672492232 4055353 4942 1225616 0.8124712787 404340010181 4944 726768 0.8416485436 9767120689 4946 1068336 0.8805527859 4180667681 4948 1603152 0.8345031758 1279124270959 4950 747450 0.8071641250 1159704007081 4952 891360 0.8149740548 1364101967 4954 1733900 0.8393620485 3501849078881 4956 594720 0.7723423113 16385484361 4958 1388240 0.9618399154 50569228469 4960 813440 0.8680593904 3930037487 4962 630174 0.8888727624 992107469 4964 1454452 0.8388185803 820409883907 4966 1455038 0.8268943841 1161574071641 4968 298080 0.8742764448 2054821 4970 685860 0.8377735104 3863252677 4972 1014288 0.8786013888 7231270463 4974 1218630 0.8120592137 11847620254121 4976 1771456 0.8635599699 3093357916003 4978 1762212 0.8347191886 11084980562807 4980 806760 0.7651954247 2047787890429 4982 1509546 0.8050403273 1443995177347 4984 1046640 0.8968469262 16174195679 4986 917424 0.7982685913 276068375017 4988 1281916 0.8583537673 87830579833 4990 1347300 0.7979239899 4407482390587 4992 524160 0.8022670052 905148287 4994 1812822 0.8971218655 9687146951987 4996 1533772 0.8372621149 582820048477 4998 324870 0.8595695977 18853409 5000 1365000 0.8722366178 1407287251891 5002 750300 0.8398345295 238656883 5004 415332 0.8897546225 19150451 [/CODE] |
[QUOTE=mart_r;612010]I'd like to take the search for T(38,16) in A[OEIS]086153[/OEIS] up to 10^16,[/QUOTE]
No solution for T(38,16) for p < 10^16. Meh. Oh well... Anyone else holding their breath for the 3rd season of "The Owl House"? |
No. I have never heard of that show.
|
[QUOTE=mart_r;596734]As a by-product, a puzzle:
Given x, find the next three consecutive primes >= x. Denote the two gaps between them g[SUB]1[/SUB] and g[SUB]2[/SUB], and let g[SUB]1[/SUB] >= g[SUB]2[/SUB]. Let r = g[SUB]1[/SUB]/g[SUB]2[/SUB]. As x becomes larger, the geometric mean r[SUB]gm[/SUB] of values of r also become larger. Find an asymptotic function f(x) ~ r[SUB]gm[/SUB].[/QUOTE] What I have so far (without explicitly claiming to be correct): Using a Poisson distributed random model, i.e. with random variables 0 < x[SUB]n[/SUB] < 1 turned into a function equivalent to the merit m[SUB]n[/SUB] = -log(1-x[SUB]n[/SUB]), we want two consecutive values m[SUB]1[/SUB] and m[SUB]2[/SUB] such that m[SUB]1[/SUB] >= m[SUB]2[/SUB] and then the geometric mean r[SUB]gm[/SUB] of (m[SUB]1[/SUB], m[SUB]2[/SUB]). For given x[SUB]2[/SUB], we use a function f(x[SUB]2[/SUB]) that gives the geometric mean of m[SUB]1[/SUB]/m[SUB]2[/SUB] for all m[SUB]1[/SUB] >= m[SUB]2[/SUB]. We have [$]\log (f(x_2)) = \frac{\int_{x_2}^1 \log(-\log(1-y)) \: \text{d}y}{1-x_2} - \log(-\log(1-x_2)).[/$] Taking all x[SUB]2[/SUB] into account, we would get [$]\log(r_{gm})=2 \cdot \int_0^{1-\varepsilon} (1-z) \log(f(z)) \: \text{d}z[/$] and lim r[SUB]gm[/SUB] = 4 for [$]\varepsilon \to[/$] 0 - [I]numerically[/I]. I don't yet know how to prove that mathematically, but as I said, I'm sure the tools are available and I leave that as an exercise for those who are more comfortable working with integrals as I am. [QUOTE=Bobby Jacobs;613597]No. I have never heard of that show.[/QUOTE] To me it's one of the best Disney shows I've seen since Chip'n'Dale's Rescue Rangers in the early 90s. I don't use streaming services, but I would be curious whether it's available on any of the popular platforms, and unabridged at that. I know that it's at least partially censored in some countries... but I don't want to spoiler anything :smile: Just, if you do, be careful to watch it in chronological order since it follows a single story plot. |
Where's the revolution? C'mon, people, you're letting me down!
The "puzzle" with the geometric mean of consecutive gaps can be generalized: for n consecutive gaps the average ratio of the largest gap divided by the smallest gap appears to be as follows (rounded to three decimal places for n>=4):
[CODE] n r_gm 1 1 2 4 3 8 4 12.641 5 17.757 6 23.249 7 29.052 8 35.121 9 41.423 10 47.928[/CODE] I'm afraid these numbers will give me headaches. 1.8 n (0.38 + log n) is an asymptotic facsimile for n<=1000, but we want more than this. What is it that I should ask myself? |
r_gm @ n=3 = 12.64200 ± 0.000015
1 Attachment(s)
All work and no pay makes me wish life wouldn't be so dull.
Here, I give you the first occurrence gaps for k=1000 up to p=2*10^14. You'll find me in the kitchen. [SIZE="1"]PS: To this day I've never expressed my continual deep appreciation for the brilliantly derived [URL="https://www.mersenneforum.org/showpost.php?p=188795&postcount=21"]joke[/URL] that came via [URL="https://www.mersenneforum.org/showpost.php?p=188790&postcount=20"]that[/URL] calculation from [URL="https://www.mersenneforum.org/showpost.php?p=188709&postcount=14"]this[/URL] old idea of mine.[/SIZE] |
I stand corrected
[QUOTE=mart_r;613861]r_gm @ n=3 = 12.64200 ± 0.000015[/QUOTE]
[YOUTUBE]yhugRD_XngM[/YOUTUBE] While I'm at it, here are the values up to n=13, corrected to be within +/- about 2 sigma: [CODE] 1 1 2 4 3 8 4 12.642007 ± 0.000009 5 17.75797 ± 0.00002 6 23.24943 ± 0.00003 7 29.05309 ± 0.00003 8 35.12109 ± 0.00004 9 41.42190 ± 0.00004 10 47.92674 ± 0.00005 11 54.62285 ± 0.00005 12 61.47478 ± 0.00006 13 68.49414 ± 0.00006[/CODE] |
[QUOTE=mart_r;613861]All work and no pay makes me wish life wouldn't be so dull.
[/QUOTE] You should get paid for this. |
[QUOTE=Bobby Jacobs;614723]You should get paid for this.[/QUOTE]
For that comment? Yeah, that was a tremendous outburst of creativity possibly worthy of a Pulitzer :wink: But you lifted my mood, so I give a small update. Prime scarcities with CSG > 1 are hard to find these days, but recently I got p[SUB]n[/SUB] = 205,465,264,987,331 which has CSG = 1.0024950 for k = 927, CSG = 1.0028949 for k = 941, and CSG = 1.0063712 for k = 939. |
You should get paid for all of the work you do in finding prime numbers.
|
The replies that I get mean more to me than any amount of money (which is too tight to mention anyway :smile:).
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Eins Zwei Drei Vier Fünf Sechs Sieben Acht
Not a record in terms of CSG, but a close contender - and the largest p for which a CSG (or "scarcity/paucity ratio") > 1 is known:
A region with exceptionally few k+1 consecutive primes for k=274..380 (listed only for CSG > 1) [CODE]k k-gap p start CSG 274 12914 309292876045019 1.00720962 275 12950 309292876045019 1.00580100 276 12996 309292876045253 1.00890285 277 13050 309292876045019 1.01561044 278 13110 309292876045019 1.02503096 279 13156 309292876045093 1.02813121 280 13230 309292876045019 1.04390467 281 13244 309292876045019 1.03252630 282 13300 309292876044949 1.04014280 283 13318 309292876044931 1.03061882 284 13342 309292876044907 1.02384129 285 13398 309292876044731 1.03143202 286 13438 309292876044811 1.03184383 287 13518 309292876044731 1.05023223 288 13542 309292876044707 1.04343738 289 13566 309292876044683 1.03667817 290 13580 309292876044683 1.02549729 291 13590 309292876044673 1.01261680 292 13620 309292876044629 1.00864965 293 13662 309292876044731 1.00999419 294 13700 309292876044731 1.00957620 295 13728 309292876044731 1.00476373 313 14448 309292876045019 1.01377017 314 14462 309292876045019 1.00309794 315 14518 309292876044949 1.01051652 316 14552 309292876045019 1.00847219 317 14604 309292876045019 1.01416841 318 14634 309292876045019 1.01041245 319 14674 309292876044949 1.01095647 320 14736 309292876044731 1.02093989 321 14760 309292876044707 1.01462007 322 14784 309292876044683 1.00833143 323 14840 309292876044731 1.01572021 324 14892 309292876044731 1.02140336 325 14922 309292876044731 1.01768328 326 14960 309292876044731 1.01738801 327 14984 309292876044707 1.01114162 328 15008 309292876044683 1.00492570 331 15134 309292876045019 1.00919910 332 15180 309292876045253 1.01231622 333 15220 309292876045213 1.01289750 334 15256 309292876045177 1.01179308 335 15340 309292876045093 1.03098513 336 15414 309292876045019 1.04600439 337 15432 309292876045019 1.03721965 338 15484 309292876044949 1.04288334 339 15502 309292876044931 1.03413607 340 15526 309292876044907 1.02796453 341 15582 309292876044851 1.03529999 342 15622 309292876044811 1.03588507 343 15702 309292876044731 1.05337889 344 15726 309292876044707 1.04718529 345 15750 309292876044683 1.04102057 346 15768 309292876044683 1.03236619 347 15778 309292876044673 1.02041740 348 15804 309292876044629 1.01519709 349 15832 309292876044601 1.01083020 350 15850 309292876044601 1.00234451 351 15906 309292876044527 1.00959606 352 15924 309292876044527 1.00113860 379 16980 309292876043453 1.00929898 380 16998 309292876043453 1.00114922 [/CODE] |
New year, new exceptional gaps
New exceptional gaps, for p = 343,408,238,858,639 each:
[CODE]k gap CSG 254 12174 1.0071454 255 12230 1.0147599 256 12248 1.0048780 (also for p-18)[/CODE] [QUOTE=mart_r;613704]The "puzzle" with the geometric mean of consecutive gaps can be generalized: for n consecutive gaps the average ratio of the largest gap divided by the smallest gap appears to be as follows (rounded to three decimal places for n>=4): [CODE] n r_gm 1 1 2 4 3 8 4 12.642 5 17.758 6 23.249 7 29.053 8 35.121 9 41.422 10 47.927[/CODE] I'm afraid these numbers will give me headaches. 1.8 n (0.38 + log n) is an asymptotic facsimile for n<=1000, but we want more than this. [/QUOTE] Since those numbers are taken from the underlying distribution process, it may also apply to primes in any admissible residue class r mod q. As a heuristical reality check, here's a sample of 30 consecutive primes congruent to 7 mod 1983: [CODE]prime; mod 1983=7 gap ratio ratio ratio ratio 10^2023+24724407 2 gaps 3 gaps 4 gaps 5 gaps 10^2023+25533471 809064 max/min max/min max/min max/min 10^2023+32002017 6468546 7.99510 10^2023+47417859 15415842 2.38320 19.0539 10^2023+53751561 6333702 2.43394 2.43394 19.0539 10^2023+59791779 6040218 1.04859 2.55220 2.55220 19.0539 10^2023+95803059 36011280 5.96192 5.96192 5.96192 5.96192 10^2023+95870481 67422 534.118 534.118 534.118 534.118 10^2023+106816641 10946160 162.353 534.118 534.118 534.118 10^2023+110156013 3339372 3.27791 162.353 534.118 534.118 10^2023+116216061 6060048 1.81473 3.27791 162.353 534.118 10^2023+117687447 1471386 4.11860 4.11860 7.43935 162.353 10^2023+126531627 8844180 6.01078 6.01078 6.01078 7.43935 10^2023+141193929 14662302 1.65785 9.96496 9.96496 9.96496 10^2023+145504971 4311042 3.40110 3.40110 9.96496 9.96496 10^2023+146421117 916146 4.70563 16.0043 16.0043 16.0043 10^2023+147230181 809064 1.13235 5.32843 18.1225 18.1225 10^2023+151517427 4287246 5.29902 5.29902 5.32843 18.1225 10^2023+154829037 3311610 1.29461 5.29902 5.29902 5.32843 10^2023+161190501 6361464 1.92096 1.92096 7.86275 7.86275 10^2023+167615421 6424920 1.00998 1.94012 1.94012 7.94118 10^2023+171010317 3394896 1.89252 1.89252 1.94012 1.94012 10^2023+172049409 1039092 3.26718 6.18321 6.18321 6.18321 10^2023+172398417 349008 2.97727 9.72727 18.4091 18.4091 10^2023+174893031 2494614 7.14773 7.14773 9.72727 18.4091 10^2023+177062433 2169402 1.14991 7.14773 7.14773 9.72727 10^2023+177895293 832860 2.60476 2.99524 7.14773 7.14773 10^2023+182031831 4136538 4.96667 4.96667 4.96667 11.8523 10^2023+190796691 8764860 2.11889 10.5238 10.5238 10.5238 10^2023+197411979 6615288 1.32494 2.11889 10.5238 10.5238 geometric mean: 3.70023 7.86887 13.4903 20.6196[/CODE] This can be generalized further to gaps between non-consecutive primes as well - either dependent on one another when running through consecutive primes (e.g. in the case k=2: 3-7, 5-11, 7-13, 11-17 etc.) or independent (by taking the differences at every other prime like 3-7, 7-13, 13-19, 19-29 etc.). Eventually these numbers appear for k=2: [CODE] r_gm r_gm (geometric mean of ratio of maximal vs. minimal gap of n consecutive gaps between a prime and the k'th next prime (here k=2), limits as prime --> oo) n dep. indep. 1 1 1 2 1.85 2.43 3 2.88 3.77 4 3.85 5.05 5 4.82 6.25 6 5.76 7.40 7 6.69 8.50 8 7.59 9.56 9 8.47 10.58 10 9.33 11.58 11 10.17 12.54 12 11.00 13.47 13 11.81 14.39[/CODE] By now I think I've lost my audience for good... [COLOR="White"](not to mention my mind ;)[/COLOR] |
You have not lost me. It is interesting that the first 3 numbers are integers:1, 4, 8. However, the next numbers are not integers. I wonder why that is.
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I'm star walkin'
1 Attachment(s)
[QUOTE=Bobby Jacobs;622006]You have not lost me. It is interesting that the first 3 numbers are integers:1, 4, 8. However, the next numbers are not integers. I wonder why that is.[/QUOTE]
You're truly my most loyal follower in this thread! To be honest, I'm not entirely sure it's exactly 8 at n=3, rather 8.000000 ± 0.000007, ballpark. I still don't know how to pin the numbers down exactly. I'm not so much wondering why the first few numbers are integers, there are a lot of sequences that start out with integers before fractional or even irrational numbers appear. What concerns me more is that I wasn't able to find a satisfying asymptotic approximation formula for larger n. As a follow-up to [URL="https://www.mersenneforum.org/showpost.php?p=609511&postcount=55"]post # 55[/URL], I've attached the data for 2*10^14 < p < 3.33333333333333*10^14, k <= 109. Meanwhile I've also looked for T(38,16) up to 2*10^16 - assuming the attached list in post # 58 is complete -, to no avail. |
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