20210714, 12:52  #12 
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
2^{3}×761 Posts 
With a LOW expectation of finding any missed primes. Error rate in LL first test is averaging ~12% per test depending partly on whether Jacobi check was included; error rate in PRP first test with GEC is probably <1ppm. Also a remarkably high number of 8digitexponent Mersenne primes were already found. If they've already been all found, we'll find no more during DC, TC, etc. Through most of recorded history, the number known lay below the cyan line representing the LenstraPomeranceWagstaff heuristic. https://www.mersenneforum.org/showpo...9&postcount=13. There's historical precedent for gaps of over 4:1 on exponent, and delays between successive discoveries exceeding a century. At our recent rate of progress of ~6M/year, a 4.1:1 gap from M51 in December 2018 would be ~338M, ~mid 2061. (Not a prediction, just a computation for comparison.)
Last fiddled with by kriesel on 20210714 at 13:13 
20210714, 19:08  #13  
"刀比日"
May 2018
13×19 Posts 
Quote:
However, the GIMPS discoveries of the last 12 Mersenne primes from M40 to M51 are hard to ignore. One should at least entertain the idea that the Mersenne primes have their own distribution which is nonlinear and so far looks like two lines with distinct slopes. Let's assume that there are several Mersenne primes within the local 9digit exponent range. Then even if there is at least one Mp with a digit sum = 35, 40,..., it is a matter of choice to ignore such digit sums (or rarely select them) and aim at digit sums that are already a part of the digit sum histogram for 51 known Mersenne primes. The prime number theorem does not tell us whether there are infinitely many Mersenne primes or not. There is an alternative for the slope of the tangent line to eventually become closer to zero and the saturated graph to end up with the last Mersenne prime. 

20210714, 19:15  #14  
Undefined
"The unspeakable one"
Jun 2006
My evil lair
18B4_{16} Posts 
Quote:


20210714, 20:03  #15  
"刀比日"
May 2018
13×19 Posts 
Quote:
I am still figuring out to what extent the combination of digit sum histograms obtained for multiple prime bases could help. 

20210714, 23:06  #16  
Apr 2020
11·53 Posts 
Quote:
Of course the PNT doesn't tell us there are infinitely many Mersenne primes; if it did, it wouldn't be an open problem. But we in fact find many more Mersenne primes than the PNT alone would suggest. It would take a significant "phase transition" for them to stop appearing  or indeed for the slope to change suddenly. I don't know of any similar examples in number theory; if you can point me to one then please do so. (Phase transitions are common in combinatorics, but in that case we generally know that there must be some change in behaviour. For example, in the ErdosRenyi random graph model, it's clear that for very small p the connected components will almost surely be very small, and for large p the graph will be almost surely connected. It's not surprising that we get a phase transition somewhere in between.) All of your hypotheses are based entirely on small patterns that are not so strange that we can discount the possibility that they are due to chance alone. I'll finish by noting that 24 out of the 51 known Mersenne primes are the first with their given exponent digit sum, so if we followed your strategy we'd miss a lot of primes. 

20210715, 02:34  #17  
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
17C8_{16} Posts 
Quote:
It seems to accommodate base 10 and 16 with the small sample sizes. Some counterexamples from base 30: Mp28, 55=5*11 Mp30, 70=2*5*7 Mp32, 84=2^{2}*3*7 Mp34, 57=3*19 Mp38, 65=5*13 Mp45, 69=3*23 Mp49, 51=3*17 Modifying it to allow for these also, to primes or products of powers of (up to 3?) small primes reminds me of Ptolemaic epicycles. Quote:
Mp51* has the first occurrence of digit sum 47 in base 10, 68 in base 16, and 108 in base 30. Testing exponents only if they had digit sums appearing for the first 50 Mersenne primes would miss it. Also Mp43, Mp39, Mp32, Mp27, Mp25, Mp24, Mp22, Mp21, Mp19, Mp15, Mp7, Mp15 produce first appearances in all 3 bases. A more productive if selfish way might be to select p = 1 mod 4, or 1 mod 8. That leaves the less likely exponents for others to process. https://www.mersenneforum.org/showpo...00&postcount=4 That approach is backed by heuristic arguments in support of Mersenne numbers with exponent = 1 mod 8 having fewer factors. There have been hundreds of documented attempts to predict or guess exponents of Mersenne primes. Also attempts made in employing base 12. No one has been successful yet. https://www.mersenneforum.org/showpo...04&postcount=5 Unique values, of sum of decimal digits of exponents of Mersenne primes currently known to Terrans, in sorted order: 2, 3, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 23, 25, 26, 28, 29, 31, 32, 37, 38, 41, 47 For base ten: Leftmost digit will be >0 so contribute at least 1, up to 9, on average 5, if uniformly distributed. Middle digits will be distributed 09 so ~4.5 each. Rightmost digit is restricted for multidigit primes to be 1, 3, 7 or 9, so ~5. The more digits, the bigger the total, on average. We're now in 9digit exponent territory, so 5 + 4.5*7 + 5 ~ 41.5 is what we might expect for future discoveries. Larger exponents are less likely to lead to primes, per candidate. That shifts the average leading digits downward somewhat, moving the expected average toward or below 41. Larger exponents are far more computationally expensive; ~tripling the exponent makes a primality test a ~10fold larger investment of computing time. That is a much larger effect on discovery probability per unit of computing time resource (GHzdecade, or some such). I think RDS would dismiss the whole thread as an exercise in numerology. Last fiddled with by kriesel on 20210715 at 03:32 

20210715, 03:17  #18 
Jun 2015
Vallejo, CA/.
3·349 Posts 
Base 10 is an eminent human construct. The only thing interesting about the number TEN (beyond us having ten fingers and 10 toes) is that 10 is both a triangular number and a tetrahedral number.
I believe that other bases like 2 should be considered. Or 12, 16, 100, 360. Perhaps pick base 88 (the number of Keys in a Grand Piano) or 57 (for the 57 varieties of the Heinz Sauce.) Last fiddled with by rudy235 on 20210715 at 03:18 Reason: an extra comma 
20210715, 03:38  #19 
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
2^{3}·761 Posts 
Bases with multiple unique small prime factors, and bases small enough for unique digit symbols in ASCII are appealing. Up to 62 is accommodated by 09AZaz.

20210715, 05:45  #20  
"Tucker Kao"
Jan 2020
Head Base M168202123
2^{5}×3×7 Posts 
Quote:
I've used this type of methods to guess on Mersenne exponents also such as: [dozenal] Z484Ӿ9277 which is [decimal] M168433723 [dozenal] (4 + 8) + (4 + Ӿ) + (9 + 2) + (7 + 7) = (10 + 12) + (Ɛ + 12) = 22 + 21 = 43 = 15 * 3 Last fiddled with by tuckerkao on 20210715 at 05:59 

20210715, 11:48  #21  
"刀比日"
May 2018
13×19 Posts 
Quote:
This thread is about paths less traveled. It is a gray area, there are no definite answers as more factual information has to be obtained both theoretically and computationally. Concerning the term 'saturation', it happens when the growth slows down and eventually stops. A common example is the logistic function, see <https://en.wikipedia.org/wiki/Logistic_function>. If selecting mainly digit sums from the histogram of known Mersenne primes, obviously one would miss the discovery of Mersenne primes with digit sums observed for the first time. However, this increases the chances of discovering Mersenne primes with digits sums most frequently occurring in the digit sum histogram of the known Mersenne primes. It is a tradeoff between the risks one would take with limited resources for a prolonged period of time. It is not like a game of poker when the results are known immediately. Analyzing incomplete patterns is better than doing nothing. The aim is at eventually assisting the selfless volunteers who run a test for a 9digit exponent for 56 consecutive years or more in a slow computer at home. If this thread is given a chance to grow and mature, there will be more clarity and understanding whether we are dealing with a mere chance or there is more to this than meets the eye. 

20210715, 12:29  #22  
"刀比日"
May 2018
F7_{16} Posts 
Quote:
The digit sum histogram is just one partial pattern characteristic among many alternative ways to analyze the known exponents. If the forum decides to keep this thread, I may post further developments in the coming years (or decades). We are not in a hurry after all. I myself have just 5 modest computers and still manage to find my user name at <https://www.mersenne.org/report_top_500>. As a volunteer, I am paying my electricity bills, and would like to increase the efficiency of my limited manual testing. In my opinion, further sophistication aided by machine learning with the use of multiple incomplete patterns could reduce the element of chance in the manual selection of exponents. 

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