New Mersenne Prime discovered (probably)

Why don't they say what it is?

Or even the number of digits.

Because the EFF Cooperative Computing Awards for the first discovery of large enough prime numbers are still active. Publishing the probable prime in advance would risk someone verifying faster than GIMPS.

It's hard to see how that someone would count as the discoverer.

That someone will totally count as the discoverer under the current rules [1], because it requires the deterministic primality proof for given number. It doesn't matter how much effort was taken to reach that candidate so far, even though it would be virtually impossible to find any new prime without that. I think the temporary embargo is fully justified for this reason.

(And I think it is technically possible to probe the reports to find what it was anyway, but it is not easy to find one at least for me. If you are really looking for that, look for the P-PRP result type.)

[1] https://www.eff.org/awards/coop/rules

How hard?

NP hard?

I wonder why?

:)

Is there a probabilistic test for Mersene primes? I thought they just wanted confirmation via independent calculation.

I used to run the Cooperative Computing Awards, and I don't think this is the reason in this case.

The most recent award was given out in 2009 for a prime over 10,000,000 digits in length. The next available award is for a prime over 100,000,000 digits in length.

But the most recent discovery by GIMPS prior to the current discovery was a prime with length only 24,862,048.

https://en.wikipedia.org/wiki/List_of_Mersenne_primes_and_pe...

The primes they've found have been getting longer by only single millions of digits every several years, so it's not very plausible that this discovery would qualify for a monetary award.

I suspect they just don't want to announce a number before it's verified on general scientific-integrity grounds.

Any probabilistic primality test will work, but GIMPS currently uses the first-time Fermat probable prime test with a very robust certificate [1] to filter almost all non-primes in advance.

[1] https://www.mersenne.org/various/math.php#prp

GIMPS currently searches for exponents up to 999,999,999, corresponding to 301,029,996 decimal digits. We don't really know much about the exact distribution of Mersenne primes so it is possible that a new discovery was from much higher ranges and thus eligible for prizes.

But yeah, they'd probably embargoed even without any potential monetary prizes because it's wise to do so in general ;-)

Sure, but all actual historical discoveries of Mersenne primes since the 1980s have been in strictly increasing size order, with no missed primes in between found in retrospect, and the successful exponents have increased gradually rather than sharply in size. It would really buck the trend to an extreme degree if the new successful exponent were 5× as large rather than something like 1.05× as large as the previous record.

I want to make an analogy to sports records, but the analogy will obviously be imperfect because the limits of human physiology are better understood in some ways than the behavior of Mersenne primes and perfect numbers. But it might be like if we heard that the marathon record had been beaten and then it turned out that the new record was something like 1:30:00 instead of something like 2:00:00. Obviously the exact value of the new record is totally unpredictable, but the best bet is that something like long-term trend lines will continue to be followed, rather than abruptly radically changed by multiple orders of magnitude.

M43112609 (2008-08) was discovered prior to M42643801 (2009-06), so that order is not really strict. I agree that there is a human tendency to test smaller primes (thus faster to verify) first, but it should be also noted that every single Mersenne number smaller than M124399361 has been already tested at least once by someone even though that limit would be way higher than what we have for primes. There are also some groups of people that specifically look for prime numbers that are barely enough to be 100,000,000+ decimal digits [1]. Given we didn't see any new Mersenne prime for many years, new prime from a random range seems much more likely than ever.

[1] See https://www.mersenne.org/primenet/ and scroll down to the starting exponent of 332,000,000. There would be an unusually large number of assigned LL/PRP tasks around this range. In fact, this holds for virtually all available PrimeNet statuses in the Wayback machine!

Sports analogies are good, but:

Any% (Anything goes to get to the "end") Video game speedruns may be ideal - a shortcut can always be found, used by everyone to quickly become zero sum + 1, then the equilibrium re-approaches optimization; but on average, gets harder and harder, as the shortcuts take skill/power/time. It also is hard to do, and easy(ish) to verify.

For linear things that hardly have any variance, you can look at longest lifespans of humans. Notably; where living 18 months longer than the next person statistically makes it more likely that you are actually your own mother and lied.

How do you prove that you verified that a number is a prime?

If you want to prove that a number is not a prime you can show the factorization or that it breack the little theorem of Fermat with 367984321568, and everyone can check the refutation inmediately.

I don't know a similar method to show that you actualy verified the number is a prime.

There are primality proving certificates.

https://en.wikipedia.org/wiki/Primality_certificate

These are sometimes infeasible for stupidly big numbers, but the Mersenne Primes have a specific structure that allows for simplification of the process.

It's called the "Lucas–Lehmer primality test". Without it, GIMPS probably wouldn't have found a single one of them in an acceptable amount of time. The one that was just discovered is HUGE. I had to write a program just to slow down the output enough to be able to appreciate its shear magnitude (it is still printing as I write this!).

They've now published the prime, and it's turned out to be somewhere in between our speculations (new exponent 1.65× the length of the previous exponent discovery).

https://www.mersenne.org/primes/?press=M136279841