mersenneforum.org > Math Mersenne gaps
 Register FAQ Search Today's Posts Mark Forums Read

 2022-11-03, 01:02 #1 FlaJunkie     Mar 2021 Rockledge, Sunny FL 2·19 Posts Mersenne gaps I am only a lowly math minor. But when I look at the chart of Mersenne Primes, it looks like some primes are missing in a few lower exponent areas. Have all numbers been checked in the red diamond areas I outlined? Is it possible one or two can still be in those areas?
 2022-11-03, 01:45 #2 Uncwilly 6809 > 6502     """"""""""""""""""" Aug 2003 101×103 Posts 5×7×311 Posts Everything in those ranges have been double checked with matching residues.
 2022-11-03, 01:55 #3 kriesel     "TF79LL86GIMPS96gpu17" Mar 2017 US midwest 162758 Posts On https://www.mersenne.org/report_milestones/ Progress toward next GIMPS milestones (last updated 2022-11-03 01:45:14 UTC, updates every 15 minutes) All exponents below M(62036137) have been tested and verified. All exponents below M(110212153) have been tested at least once.
 2022-11-03, 13:14 #4 ATH Einyen     Dec 2003 Denmark 2×17×101 Posts You have year of discovery on the x-axis, but that has no impact on whether or not there is a Mersenne Prime in that exponent range. Here is the current conjecture about the Mersenne Prime distribution: https://primes.utm.edu/notes/faq/NextMersenne.html Since the conjecture concerns the logarithm of the exponent you would expect roughly the same number of exponents between for example powers of 10. But there are actually surprisingly many Mersenne Primes found in the last 15-20 years between 10M and 100M: Code: Exponents Mersenne Primes 0-10 4 10-100 6 100-1000 4 1000-10000 8 10000-100000 6 100000-1000000 5 1000000-10000000 5 10000000-100000000 13
2022-11-03, 13:40   #5
Uncwilly
6809 > 6502

"""""""""""""""""""
Aug 2003
101×103 Posts

5×7×311 Posts

Quote:
 Originally Posted by ATH Here is the current conjecture about the Mersenne Prime distribution:
Conjecture being the important word there.

2022-11-03, 16:25   #6
kriesel

"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest

162758 Posts

Quote:
 This means that the geometric mean of the ratio of two successive Mersenne exponents is 2 raised to 1/egamma or about 1.47576
https://primes.utm.edu/notes/faq/NextMersenne.html
So in a power of ten range of exponents we would expect if the conjecture is right, about 1/log10(1.47576) ~5.92 Mersenne primes, and in 8 powers of 10 (100=1 to 108, which we've already tested past once), ~47.3. And note there are good reasons to expect variations from the mean asymptotic rate. Empirically, the ratio between known consecutive Mersenne primes' exponents range from ~1.015 to 4.1024. The series necessarily starts off slow with just 4 between 1 and 10.

We've already tested once up exhaustively up to 110212153, ~82589933 * 1.33445.

It's possible another relative drought lies between M82589933 and ~100Mdigit or extending a bit beyond.
It would not need to be a record long ratio for there to be none in that span; 82589933*4.1024 ~ 338816941, ~1.99% beyond 100Mdigit threshold.

Last fiddled with by kriesel on 2022-11-03 at 16:38

2022-11-03, 17:30   #7
ATH
Einyen

Dec 2003
Denmark

1101011010102 Posts

Quote:
 Originally Posted by Uncwilly Conjecture being the important word there.
Yes, definitely only a conjecture. But I meant that it does not make sense looking at blank areas on a graph, where one axis is the discovery date. That has no bearing on what Mersenne Primes exists in those ranges.

2022-11-03, 19:51   #8
FlaJunkie

Mar 2021
Rockledge, Sunny FL

2·19 Posts

Quote:
 Originally Posted by ATH Yes, definitely only a conjecture. But I meant that it does not make sense looking at blank areas on a graph, where one axis is the discovery date. That has no bearing on what Mersenne Primes exists in those ranges.

Duh, again. You are absolutely correct. I misread the graph while doing several other things. In fact, I've screwed up several questions on other parts of the forum.

And I know many people here don't like me for jumping in on that soapbox derby recently, but I really like the whole Mersenne thing...and I'm just gonna quietly back out of here for a while and let my Ryzen keep poking at the 100+ million digit numbers. Future questions and comments can wait.

Thanks for the positive responses to date. You know who you are.

 2022-12-13, 14:16 #9 Andrew Usher   Dec 2022 3068 Posts It's safe to assume the conjecture (which applies to repunits in all bases, by the way) is at least a good approximation. But there's no way to predict the next Mersenne at all, and it's pointless to try. There's been two gaps near 4:1, so it's not impossible the next one could be 100+ million digits. But that doesn't make searching there a better chance.

 Similar Threads Thread Thread Starter Forum Replies Last Post Bobby Jacobs Prime Gap Searches 52 2020-08-22 15:20 ET_ FermatSearch 59 2018-07-27 17:05 robert44444uk Prime Gap Searches 1 2018-07-10 20:50 robert44444uk Prime Gap Searches 77 2016-12-28 08:19 gd_barnes Riesel Prime Search 11 2007-06-27 04:12

All times are UTC. The time now is 15:46.

Fri Jan 27 15:46:06 UTC 2023 up 162 days, 13:14, 0 users, load averages: 1.68, 1.45, 1.28