Percent Chance of a New Mersenne Prime Being Found in 2023

[Tyler Busby]

I saw an interesting market on a (fake money) prediction market website today, and was curious if anybody here would have a methodology for estimating the percent chance of a new Mersenne prime being found in 2023. (Or any hypothetical future year)

I'd think you could take the remaining GIMPS forecasted FLOPS for the year, estimate what number range will be checked (probably easier said than done, and outside my current ability/knowledge), and then do a density of primes based calculation (or perhaps something a bit more clever), but I'm curious if any of you would be confident enough in your percent prediction to put "theoretical" money on it, or what range of probabilities you might give it.

Basically, I thought the mental exercise might be a good way for me to learn more details about GIMPS (as I have mostly been interested in factoring or other prime forms in the past)

[retina]

Originally Posted by Tyler Busby

I'd think you could take the remaining GIMPS forecasted FLOPS for the year, estimate what number range will be checked ...

Those are the easy estimates.

Originally Posted by Tyler Busby

... and then do a density of primes based calculation ...

That is the hard part. So far no knows if there are any more primes to be found. Or if someone does know, then they've kept it a secret. Maybe we have found all 51 primes and that is all there are?

[Dr Sardonicus]

The probability of another Mersenne prime being discovered in 2023 is a member of the set {0, 1}. Which member will become known by January 1, 2024. If it becomes known before January 1, 2024, it will be 1. Otherwise it will be 0.

An easier estimate is the probability that all known-composite M_p for prime p < 10000 will be completely factored by the end of 2023. I confidently predict that probability is 0.

[charybdis]

Originally Posted by Dr Sardonicus

The probability of another Mersenne prime being discovered in 2023 is a member of the set {0, 1}. Which member will become known by January 1, 2024. If it becomes known before January 1, 2024, it will be 1. Otherwise it will be 0.

Don't think so. If you toss a coin, the probability that it will come up heads is 1/2. Not "0 or 1 but we don't know which yet". That's the whole point of probability.

Yes, the exponents of the Mersenne primes are predetermined, so this is slightly different from the coin. But we don't know exactly which exponents are going to be tested this year, so even from the perspective of some entity that already knows all the Mersenne primes, there is still some uncertainty.

[ATH]

http://hoegge.dk/mersenne/GIMPSstats.html

The NO-LL column in 100M-332M went down 95433 in the last 365 days, while the 100M-332M Factored column went up with 37482, which means 95433 - 37482 = 57951 PRP tests (and a few LL tests) were done in 100M-332M in the last 365 days.

Assuming this pace continues and there are 206 days left of 2023, then 206*57951/365 ~ 32707 more exponents will be finished in 100M-332M for the rest of 2023.
There are 61704 exponents left at 112M-118M, so lets assume 116M is the average of the exponents that will be done the rest of the year.

https://t5k.org/mersenne/heuristic.html
This is heuristics and not proven, but lets assume that:
The probability that 2^p-1 is prime is about (e^gamma log (ap) )/(p log 2) where a=2 if p=3 (mod 4), and a=6 if p=1 (mod 4).

Lets use a=4 as the average, since there will be both p=1 (mod 4) and p=3 (mod 4).

e^gamma * log(4 * 116000009) / (116000009 * log 2) = 4.42*10^-7 or 1 in 2.26M

So in total for 2023: 32707 in 2.26M ~ 1 in 69 (nice )



It doesn't really mean that much, because if you had done this calculation 5-10 years ago at lower exponents, you might have gotten lets say 1 in 30 or 1 in 20 for that year, but back then it didn't take 20 or 30 years to find the next prime. In fact we found more than we expected.

[charybdis]

Originally Posted by ATH

This is heuristics and not proven, but lets assume that:
The probability that 2^p-1 is prime is about (e^gamma log (ap) )/(p log 2) where a=2 if p=3 (mod 4), and a=6 if p=1 (mod 4)...

So in total for 2023: 32707 in 2.26M ~ 1 in 69 (nice )

Ah, but you're not taking into account that the exponents being PRP-tested have no factors less than ~2^77, so are more likely to be prime than the heuristic suggests.

Really you should sum the heuristic probabilities for all exponents in the range that is being tested, but we can quickly get a good estimate: about 65% of exponents in that range have a known factor, so the candidates that make it through to PRP are more likely to be prime by a factor of 1/(1-0.65).
Change 69 to 69*(1-0.65): 1 in 24. Seems more reasonable.

[Andrew Usher]

That is exactly the right correction - there is no need to calculate exactly the primality chance for each unfactored number (as is required for cofactors) as the density of expected primes in any interval is not changed by removing some of the composites. I have done rough estimates of this before - using whole years, not partial years - and concluded the expected time to find the next prime at around 10 years (reasonably close to that).

Originally Posted by retina

That is the hard part. So far no knows if there are any more primes to be found. Or if someone does know, then they've kept it a secret. Maybe we have found all 51 primes and that is all there are?

I hope this was not entirely serious. The heuristic density of primes is not 'the hard part', it's the only serious estimate we have. Surely you can't believe we can't predict anything without rigorous proof - outside mathematics, that wouldn't work at all.

[retina]

Originally Posted by Andrew Usher

I hope this was not entirely serious. The heuristic density of primes is not 'the hard part', it's the only serious estimate we have. Surely you can't believe we can't predict anything without rigorous proof - outside mathematics, that wouldn't work at all.

If you have a proof of the infinitude of Mersenne prime please post it. I'm sure many want there to be more primes (myself included), but wanting is not knowing.

[charybdis]

Originally Posted by Andrew Usher

...there is no need to calculate exactly the primality chance for each unfactored number...

I am well aware. "All exponents in the range" = including the factored ones.
And yes, I'm also aware that one does not find the probability of a union by summing the probability of the individual events. But here it's a good enough approximation to the (conjectured) truth.

[Andrew Usher]

Well, the events are independent, so the probability of the union can be calculated exactly, and yes, here the difference between the simple sum and the disjunction may be neglected compared to other errors in the estimate.

Originally Posted by retina

If you have a proof of the infinitude of Mersenne prime please post it. I'm sure many want there to be more primes (myself included), but wanting is not knowing.

No one has such a proof, but neither does anyone have a concrete reason to doubt it - so again I don't understand what this could mean seriously. When nothing is proved, we simply must take what is most likely to be true, rather than saying all possiblities must be considered equally. We do not seriously doubt, say, Goldbach's conjecture, we do not assume that PRPs are as likely as not to be actually composite, and in other cases as well should not consider perverse distributions of primes without reason.

[retina]

Originally Posted by Andrew Usher

No one has such a proof, but neither does anyone have a concrete reason to doubt it - so again I don't understand what this could mean seriously. When nothing is proved, we simply must take what is most likely to be true, rather than saying all possiblities must be considered equally. We do not seriously doubt, say, Goldbach's conjecture, we do not assume that PRPs are as likely as not to be actually composite, and in other cases as well should not consider perverse distributions of primes without reason.

Who is "we"? Speak for yourself.

You appear to have taken it on faith that MPs will continue beyond the 51 found so far. I hope you are correct. It would be sad to have no more. But if one doesn't consider the possibility of there being no more MPs then the stats could be off. If you want to compute the stats then also include the possibility of there being no more MPs and update the results.