Triple prime search
Hi,
currently I am interested in triple primes. This is mainly because the record primes listed in Chris Caldwell's database are still quite small and therefore quick to check. While choosing a 'n' to sieve, I asked myself whether there is a list or some coordinated prime search project for triple primes where one could check if the 'n' was already searched or up to which upper limit the exponent was searched. I know that there are several similar projects / private range reservations etc. for other prime numbers. If there is no such project, would it be of interest to create one? regards hunson 
Look at [url]http://www.pzktupel.de/ktuplets.htm[/url]
4111286921397 * 2^66420 + d, d = 1, 1, 5 seeem to be the largest one know Also look at this thread [url]https://www.mersenneforum.org/showthread.php?t=16705&highlight=tuple[/url] 
There are some data at [URL="oeis.org"]oeis.org[/URL].
I did a search for 'prime triple' and received some results. These are, of coarse, only the smallest few triples of each kind. Also, there is some information in Wikipedia under[URL="https://en.wikipedia.org/wiki/Prime_ktuple"] ktuple[/URL]. Regards, Matt 
Thanks for the answers.
@firejuggler: The page you mentioned is about that what I thought about. I image a project that summarizes the ranges that where tested with the count of primes, twins, triples, sieve depth etc. so that anyone who wants to search for triple primes can easily see whether the range that he choose was already worked on. Similar to the CarolKyneaProject ([url]https://www.mersenneforum.org/showthread.php?t=21216[/url]). I have a chemistry background and me and some friends back in uni imagined how useful a journal would be that does not show the latest and greatest in organic chemistry, but the things not to try :) The title would be something like "Don't waste your time trying this ...". So in that fashion one could image a summery of the ranges searched for triple primes. This would be especially interesting for "nice" exponents like k*2^55555, k*2^54321 or palindromic exponents etc. I am certainly not the only one that finds primes with these desirable. But since there is no public documentation about the sieving / checking efforts one could easily waste time on these if they were already checked up to a certain bound. hunson 
Hi hunson, welcome!
I do hold the current triple record. It's true, these numbers aren't really that big compared to twins or single primes records. It's not my goal to discourage you from trying. These records are meant to be broken, after all. Still you need to be aware that finding a constellation of e.g. three primes is much harder than finding a single prime. But what really kept me from going higher is the fact that in a triplet there will always be one prime that can't be verified using LLR but will require a general prime proving software such as PRIMO. At 20.000+ digits you are looking at several months of computing on a powerful machine just to get that proof without knowing for sure it will be possible to prove it. There are limitations to the software that might make it impossible to prove a number prime even when it's smaller than PRIMOs current limit. Never had that happen to me though. You could also try to find numbers of a special form so you can prove them thanks to knowing enough "helper primes" but that will slow down the sieving process. I'll be happy to point you in the right direction if you want to go this way. There's information and software available in this forum. I'll have to look for the right threads if you like. TL;DR: I have considerable computing power and after doing trial runs & calculations I decided to not spend years trying to go higher. But that's just me. If you want to try, more power to you. Just be aware of the amount of work you're taking on. If you find enough people to help, it might be a fun project to run. 
Thanks for your input PuzzlePeter.
I do not shoot for a new record prime, I do not have the computing power for this. I am very pleased with a top 10, like with my twinprime which was on the 9th place at the time. This is all just a fun project for me. I am aware that I have to check the +5 form with PRIMO, that's mainly the reason why a ~12700 digit number is my goal. It should be still manageable to check this within a month or so on reasonable new hardware I guess. The idea for a project page with an overview of the checked ranges/exponents might be still useful for someone who wants to find a prime in the top 15 like me!? [QUOTE=PuzzlePeter;593091] You could also try to find numbers of a special form so you can prove them thanks to knowing enough "helper primes" but that will slow down the sieving process. I'll be happy to point you in the right direction if you want to go this way. There's information and software available in this forum. I'll have to look for the right threads if you like. [/QUOTE] Any help is very much appreciated! What forms do you suggest? 
[QUOTE=PuzzlePeter;593091]Hi hunson, welcome!
I do hold the current triple record. It's true, these numbers aren't really that big compared to twins or single primes records. It's not my goal to discourage you from trying. These records are meant to be broken, after all. Still you need to be aware that finding a constellation of e.g. three primes is much harder than finding a single prime. But what really kept me from going higher is the fact that in a triplet there will always be one prime that can't be verified using LLR but will require a general prime proving software such as PRIMO. At 20.000+ digits you are looking at several months of computing on a powerful machine just to get that proof without knowing for sure it will be possible to prove it. There are limitations to the software that might make it impossible to prove a number prime even when it's smaller than PRIMOs current limit. Never had that happen to me though. You could also try to find numbers of a special form so you can prove them thanks to knowing enough "helper primes" but that will slow down the sieving process. I'll be happy to point you in the right direction if you want to go this way. There's information and software available in this forum. I'll have to look for the right threads if you like. TL;DR: I have considerable computing power and after doing trial runs & calculations I decided to not spend years trying to go higher. But that's just me. If you want to try, more power to you. Just be aware of the amount of work you're taking on. If you find enough people to help, it might be a fun project to run.[/QUOTE] Didn't you have a form that could be proved for all of 1/+1/+5 that could be sieved using polysieve? 
[QUOTE=henryzz;593134]Didn't you have a form that could be proved for all of 1/+1/+5 that could be sieved using polysieve?[/QUOTE]
Yes, I just needed some time to hunt down the thread. It starts around here: [URL]https://mersenneforum.org/showthread.php?t=16705&page=8[/URL] In short: The idea was to start with a pair of precomputed twins. I'll call them s and s+2. These will be of the form k*b^n+d. I then wanted to use the following formula for the candidates: N=(s+2)*(a*s^23) + {1, +1, +5} with "a" being the running variable. N=a*s^33*s+2*a*s^26 Sorting into terms with and without "a": N=(63*s)+a*(2*s^2+s^3) + {1, +1, +5} N=(s+2)*(a*s^23)1 can be tested using N+1 test and known prime factor s+2 N=(s+2)*(a*s^23)+1 can be tested using N1 test and known prime factor s+2 N=(s+2)*(a*s^23)+5 =a*s^33*s+2*a*s^26 +5 = a*s^33*s+2*a*s^21 can be tested using N+1 test and known prime factor s N=(s+2)*(a*s^23)+7 =a*s^33*s+2*a*s^26+7 = a*s^33*s+2*a*s^2+1 can be tested using N1 test and known prime factor s which would give a quadruplet So for triplets and quadruplets every number can be proven using one of the twin primes as a helper and the software PFGW. But how to do the sieving? I am a really bad programmer, but Robert Gerbicz was so nice and wrote his software "Polysieve" Polysieve uses the form sorted into terms with and without "a" and calls them P(s) and a*Q(s) respectively. So for the example above we get P(s)=62*s and Q(s)=3*s^2+s^3 Polysieve will ask for the degrees of P and Q, the coefficients and the "c" values (1, +1, +5 in our example for a triplet, additionally +7 for quadruplet) and after sieving to your chosen limit will write a candidate file in ABC format that can be tested using PFGW. There will probably be some questions left. Just ask, here or in PM. In case the "Germany" under your name means that German is your native language  so is mine. That might make the PMs a lot easier :smile: 
[QUOTE=henryzz;593134]Didn't you have a form that could be proved for all of 1/+1/+5 that could be sieved using polysieve?[/QUOTE]
Nice, not remembered for this, but for "large" n the k*2^n1+1,+5 formula shouldn't be winning due to the slightly faster prp tests on special form? You would need T(N)=O(log(N)^5/loglog(N)^2) bit operations (including sieve) to find a prp triplet near N=2^n, but for large N the cost of the "slow" primality test for the ugly k*2^n+5 number using elliptic curves (say with Primo) is still faster than T(N). 
[QUOTE=R. Gerbicz;593172]Nice, not remembered for this, but for "large" n the k*2^n1+1,+5 formula shouldn't be winning due to the slightly faster prp tests on special form?
You would need T(N)=O(log(N)^5/loglog(N)^2) bit operations (including sieve) to find a prp triplet near N=2^n, but for large N the cost of the "slow" primality test for the ugly k*2^n+5 number using elliptic curves (say with Primo) is still faster than T(N).[/QUOTE] I'd never doubt your calculations. But the N ranges I am interested in are larger than the Primo limit. So "the polysieve way" is the only one I know to get large provable triplets or quadruplets. 
Hi again all,
Here are two links that seem to be germane to this discussion. Regarding 3 primes with the pattern (1, 1, 5). [URL="https://primes.utm.edu/glossary/xpage/PrimeTriple.html"]https://primes.utm.edu/glossary/xpage/PrimeTriple.html [/URL] [URL="http://www.pie.de/ktuplets.htm#largest3"]http://www.pie.de/ktuplets.htm#largest3 [/URL] Also, the first few small ones are in OEIS.org at [URL="http://oeis.org/A007529"]A007529[/URL] Hope that helps. Regards, Matt 
I made a longish list of prime triple patterns or 3tuples already in the database.
Some entries from The Online Encyclopedia of Integer Sequences A022004 Initial members of prime triples (p, p+2, p+6). 5,11,17,41,101,107 and so on A046134 p, p+2 and p+8 are primes. 3, 5, 11, 29, 59, 71, 101, 149, 191, 269 and so on A046135 Primes p such that p+2 and p+12 are primes. 5, 11, 17, 29, 41, 59, 71, 101, 137, 179, 227, 239, 269, 281, 347, 419 and so on A046136 Primes p such that p, p+4 and p+10 are primes. 3, 7, 13, 19, 37, 43, 79, 97, 103, 127, 163, 223, 229, 307, 349, 379, 439 and so on A046137 Primes p such that p+4 and p+12 are also prime. 7, 19, 67, 97, 127, 229, 397, 487, 739, 757, 907, 1009 and so on A046138 p, p+6 and p+8 are primes. 5, 11, 23, 53, 101, 131, 173, 191, 233, 263, 563, 593, 653 and so on A046139 p, p+6 and p+10 are primes. A046141 p, p+8 and p+12 are primes. I have tried to contribute a few prime triples to OEIS but was told that these sequences are 'not of general interest'. Maybe someone else would like to try. Note the pattern Primes p such that p, p+2 and p+14 is not in the OEIS database. It starts with 3,5,17,29,59,137,149,179,197,227,269,419,599,617,659,809,1019,1049,1277,1289,1607,1787,1997 and so on. Any takers? Regards, Matt 
triple prime log
Again,
If someone wants to put a few prime triples in OEIS, you can try. I also tried (0, 2, 20) but that one was rejected too. "not of general interest". But I have made donations to OEIS and I received a holiday card from Neil Sloan himself, so the wheel may be greased. Regards, Matt 
a short calculation
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Now I have a calculation to share. It is a 3tuple. Pattern is (0,2,18). I 'dialed it in' to fit on one piece of paper, and printed one. I used Maple computer algebra system and computer language.
I bet you can't calculate more :smile: Look Matt Eddited to add  Have you ever dialed in a radio station? It is fun. Some of us know the joy of finding special numbers. You, dear reader, and I are special. We are elite. Some people can't even read. Best of luck. M 
Triple prime search with pattern (0, 2, 20)
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Where 20 is an arbitrary parameter :)
This is what I was told by an OEIS.org referee. *sigh* Hi again all, Here are more triple prime search data. It is for your enjoyment. I put it on this public forum so that it may be shared. These are prime numbers p such that p, p+2, and p+20 are all prime numbers. So, this is a pattern of (0, 2, 20). And these are the smallest such positive prime numbers. And all prime numbers are positive integers. Again, someone else can try to put this in OEIS. Because, hey mankind needs to know this. (But maybe, 'not of general interest') Regards, Matt :mooc: 
You could use PFGW with this script:
[code] ABC2 $a & $a+2 & $a+20 a: from 1 to 20000 step 2 [/code] This will find all triples to n=20k within a minute! The log file "pfgwprime.log" will contain all pairs in this form: [code] 3 3+2 3+20  Complete Set  [/code] 
Just to make the list a bit more exclusive, I told PariGP to look for primes p such that p + 2 and p + 18 were the [i]next two consecutive primes[/i]. With the small limit I chose, it would have taken more time figuring out time saving refinements than it took to write and run a mindless script.
[code]k=0;forprime(p=3,35000,q=nextprime(p+1);r=nextprime(q+1);if(qp==2&&rq==16,k++;print(k" "p" "q" "r))) 1 1931 1933 1949 2 2111 2113 2129 3 2591 2593 2609 4 2801 2803 2819 5 3119 3121 3137 6 3371 3373 3389 7 3389 3391 3407 8 5021 5023 5039 9 5279 5281 5297 10 5879 5881 5897 11 6761 6763 6779 12 7331 7333 7349 13 9011 9013 9029 14 9239 9241 9257 15 10271 10273 10289 16 11351 11353 11369 17 11699 11701 11717 18 16631 16633 16649 19 17579 17581 17597 20 17789 17791 17807 21 18059 18061 18077 22 18311 18313 18329 23 18521 18523 18539 24 19139 19141 19157 25 20231 20233 20249 26 20771 20773 20789 27 22091 22093 22109 28 22619 22621 22637 29 24179 24181 24197 30 26861 26863 26879 31 27281 27283 27299 32 30011 30013 30029 33 31121 31123 31139 34 32141 32143 32159 35 32939 32941 32957 36 33809 33811 33827 37 34649 34651 34667 ? [/code] 
[QUOTE=MattcAnderson;594139]Where 20 is an arbitrary parameter :)
This is what I was told by an OEIS.org referee. <snip>[/QUOTE]In the spirit of "20 is an arbitrary parameter," the following mindless script looks for a triple p, p + 2, p + k which are [i]consecutive[/i] primes, for even integers k > 2 which are not congruent to 1 (mod 3). I ran the thing out to fifty million. The triples found may not be the least for the indicated k. [code]? k=4;d=2;forprime(p=5,50000000,q=nextprime(p+1);r=nextprime(q+1);if(qp==2&&rq==k,print(k+2" "p" "q" "r);k+=d;d=6d)) 6 5 7 11 8 29 31 37 12 137 139 149 14 197 199 211 18 1931 1933 1949 20 4157 4159 4177 24 7127 7129 7151 26 12071 12073 12097 30 13337 13339 13367 32 13931 13933 13963 36 20441 20443 20477 38 28349 28351 28387 42 38459 38461 38501 44 58787 58789 58831 48 163061 163063 163109 50 172439 172441 172489 54 405089 405091 405143 56 439007 439009 439063 60 510617 510619 510677 62 667019 667021 667081 66 997811 997813 997877 68 1128821 1128823 1128889 72 1149059 1149061 1149131 74 1152419 1152421 1152493 78 1188071 1188073 1188149 80 1354391 1354393 1354471 84 1448219 1448221 1448303 86 1601867 1601869 1601953 90 3227039 3227041 3227129 92 4575377 4575379 4575469 96 4797071 4797073 4797167 98 4927931 4927933 4928029 102 5723897 5723899 5723999 104 8981459 8981461 8981563 108 12591041 12591043 12591149 110 13545797 13545799 13545907 114 15492437 15492439 15492551 116 15879047 15879049 15879163 120 16054277 16054279 16054397 122 18931697 18931699 18931819 126 36084311 36084313 36084437 128 43469651 43469653 43469779 ?[/code] For each k = 6*t + 2 or 6*t + 4 up to 150, these are k, and the [i]least[/i] triples p, p+2, p+k which are [i]consecutive[/i] primes. Note that p does not increase monotonically with k :smile: [code]6 5 7 11 8 29 31 37 12 137 139 149 14 197 199 211 18 1931 1933 1949 20 521 523 541 24 1949 1951 1973 26 1667 1669 1693 30 2969 2971 2999 32 7757 7759 7789 36 12161 12163 12197 38 28349 28351 28387 42 20807 20809 20849 44 16139 16141 16183 48 163061 163063 163109 50 86627 86629 86677 54 25469 25471 25523 56 40637 40639 40693 60 79697 79699 79757 62 149627 149629 149689 66 625697 625699 625763 68 552401 552403 552469 72 173357 173359 173429 74 360089 360091 360163 78 716171 716173 716249 80 281429 281431 281509 84 265619 265621 265703 86 637937 637939 638023 90 544277 544279 544367 92 404849 404851 404941 96 1599707 1599709 1599803 98 1242641 1242643 1242739 102 838247 838249 838349 104 8981459 8981461 8981563 108 4297091 4297093 4297199 110 3593201 3593203 3593311 114 2637797 2637799 2637911 116 4131107 4131109 4131223 120 1349531 1349533 1349651 122 1895357 1895359 1895479 126 9707987 9707989 9708113 128 5825999 5826001 5826127 132 11188757 11188759 11188889 134 29980409 29980411 29980543 138 24947189 24947191 24947327 140 11501879 11501881 11502019 144 10343759 10343761 10343903 146 25507421 25507423 25507567 150 19918751 19918753 19918901[/code] 
I appreciate all the typed comments you men.

more prime numbers  k tuples are more general than prime constellations
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Wow!
Man, Norman Luhn from Germany (we are all residents of our earth) I am impressed by all of your hard work. Now I am an old man (but less than 50 years old :) My original calculations can be found at [URL="mattanderson.fun"]mattanderson.fun[/URL] But my domain is only rented for another nine years (until 2030). I plan to rerent the domain and do more through my internet service provider. I use GoDaddy.com. I have not complaints about them, but they are expensive  $80 USD for two service calls. But worth it. We are interested in ktuples. Someone else might do more than me. Regards, Matt 
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