20221103, 01:02  #1 
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? 
20221103, 01:45  #2 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
2A8B_{16} Posts 
Everything in those ranges have been double checked with matching residues.

20221103, 01:55  #3 
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
2·29·127 Posts 
On https://www.mersenne.org/report_milestones/
Progress toward next GIMPS milestones (last updated 20221103 01:45:14 UTC, updates every 15 minutes)

20221103, 13:14  #4 
Einyen
Dec 2003
Denmark
2·17·101 Posts 
You have year of discovery on the xaxis, 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 1520 years between 10M and 100M: Code:
Exponents Mersenne Primes 010 4 10100 6 1001000 4 100010000 8 10000100000 6 1000001000000 5 100000010000000 5 10000000100000000 13 
20221103, 13:40  #5 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
10101010001011_{2} Posts 

20221103, 16:25  #6  
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
2×29×127 Posts 
Quote:
So in a power of ten range of exponents we would expect if the conjecture is right, about 1/log_{10}(1.47576) ~5.92 Mersenne primes, and in 8 powers of 10 (10^{0}=1 to 10^{8}, 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 20221103 at 16:38 

20221103, 17:30  #7 
Einyen
Dec 2003
Denmark
6552_{8} Posts 

20221103, 19:51  #8  
Mar 2021
Rockledge, Sunny FL
38_{10} Posts 
Quote:
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. 

20221213, 14:16  #9 
Dec 2022
211 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.

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