Mersenne gaps

[FlaJunkie]

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?

[Uncwilly]

Everything in those ranges have been double checked with matching residues.

[kriesel]

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.

[ATH]

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:

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

[Uncwilly]

Originally Posted by ATH

Here is the current conjecture about the Mersenne Prime distribution:

Conjecture being the important word there.

[kriesel]

This means that the geometric mean of the ratio of two successive Mersenne exponents is 2 raised to 1/e^gamma 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.

[ATH]

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.

[FlaJunkie]

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.

[Andrew Usher]

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.