[QUOTE=bloodIce;423043]Is there any other prime of a special form, which could be tested for a similar time as the most mersennes in the current wave? Just curiosity, if there is a chance that a non-mersenne prime will hold a record ever.[/QUOTE]
Yes, for example: Let m=Mp be a mersenne prime, of the form Mp=2^p-1, for a prime p (we know 49 of them). Then Mm=2^m-1 [U]might be[/U] prime, because its exponent is prime. We can never run a LL test for such a huge number (even MM127 (=\(2^{2^{127}-1}-1\)) is way out of reach!), but we can try to factor it. All its factors will be of the form q=2*k*m+1 for some small k, and if this q is smaller than (roughly) m^2, then it is prime (because a composite product of two factors, both of the same form, is bigger than m^2). This is what double-mersenne project is doing, and testing if q=2*k*m+1=2*k*(2^p-1)+1 is a factor of Mm takes about the same time as a LL test for Mp. With some luck, one can find a q=2*k*M49+1 which divides the freshly-born MM49, for a small k, and that [U]will be[/U] a record (i.e. bigger that M49). So, yes, it is "quite easy" for a record prime to be of the form 2*k*m+1, where m is a mersenne prime, and k is smaller than m. Unfortunately there is only a fistful of guys working for double mersenne project, as opposite to the thousands working for GIMPS.

[QUOTE=Madpoo;423035]The perfect # for M74207281 contains "74207281" itself, around 74% of the way in.[/QUOTE][url]http://www.mersenneforum.org/showthread.php?t=5414[/url]

[QUOTE=bloodIce;423043]Is there any other prime of a special form, which could be tested for a similar time as the most mersennes in the current wave? Just curiosity, if there is a chance that a non-mersenne prime will hold a record ever.[/QUOTE]

Numbers of the form k*b^n-1 or k*b^n+1 are a bit slower to test but it is only by a small constant term I believe.

[URL="http://www.popularmechanics.com/science/a19030/newly-discovered-prime-number-is-22-million-digits-long/"]Popular Mechanics[/URL]

Forbes and New Scientist have articles on this prime number.
[QUOTE]
[URL="http://www.forbes.com/sites/kevinknudson/2016/01/19/new-record-prime-number-discovered/#15f226d04b3c2a2289194b3c"]New Record Prime Number Discovered[/URL]
Forbes - ‎1 hour ago‎
A Mersenne prime is a prime number of the form M(n) = 2[SUP]n[/SUP] – 1 for some integer n. It's easy to see that n must be a prime number in order for M(n) to be prime and while the first few Mersenne numbers are prime the first counterexample is M(11) = 23 × 89.

[URL="https://www.newscientist.com/article/2073909-prime-number-with-22-million-digits-is-the-biggest-ever-found/p"]Prime number with 22 million digits is the biggest ever found[/URL]
New Scientist - ‎1 hour ago‎
It's time for a new prime to shine. The largest known prime number is now 2[SUP]74,207,281[/SUP] – 1, smashing the previous record by nearly 5 million digits. This mathematical monster was discovered by Curtis Cooper at the University of Central Missouri in ...[/QUOTE]

[QUOTE=henryzz;423053]Numbers of the form k*b^n-1 or k*b^n+1 are a bit slower to test but it is only by a small constant term I believe.[/QUOTE]
Henry's got it, and the size of k determines how much larger an FFT must be run than the equivalent-size Mersenne candidate.

Perusing the list of 100 largest known primes (at primes.utm.edu) shows what forms are relatively easy to test at large sizes. Mersennes are easier-than-most to prefactor deeply, allowing fewer primality tests per (expected) prime.

[QUOTE=henryzz;423053]Numbers of the form k*b^n-1 or k*b^n+1 are a bit slower to test but it is only by a small constant term I believe.[/QUOTE]

For what it's worth, Airsquirrels tested:
[CODE]Wagstaff W74207281 = (2^74207281+1)/3[/CODE]for PRP... came up negative.
_______________________

[COLOR=Blue](SB, 19-Jan:) ...and Andreas ATH attempted factoring W[/COLOR][COLOR=Blue][COLOR=Blue][SUB]74207281[/SUB][/COLOR] quite deeply with mfaktc-W before the PRP test, of course. No "small" factors are yet known for W[SUB]74207281[/SUB] (at least to 78 bits)
[/COLOR]
[COLOR=Blue](SB, 22-Jan:) Ran some ECM and a P-1 on the [/COLOR][COLOR=Blue][COLOR=Blue]W[/COLOR][COLOR=Blue][COLOR=Blue][SUB]74207281[/SUB]
[/COLOR][/COLOR]2^74207281[B]+1[/B] completed P-1, B1=500000, B2=30000000, E=12, We8: 00000000
[/COLOR]

So... did anyone manage to guess the exponent before it was announced?

[QUOTE=VBCurtis;423060]Mersenne's are easier-than-most to prefactor deeply, allowing fewer primality tests per (expected) prime.[/QUOTE]
That's exactly the important advantage of Mersenne's, as well as Generalized Fermat Numbers (GFN) and their cousins Phi(3*2^m,b) cyclotomic numbers ("Cyclo"). (There are also Gaussian-Mersenne and Eisenstein-Mersenne number sequences with the same benefit.) Because all of them have [I]restricted[/I] proper divisors, their candidate lists are [B]much[/B] more enriched than any other competitors. This allows to avoid many more costly tests than for other sequences. The second consideration is the fastest costly test; and these are almost the same for Mersenne's, GFNs and "Cyclo"s.

One open question is if Gaussian-Mersenne and Eisenstein-Mersenne numbers' test speeds can be improved in a manner demonstrated by Yves Gallot for GFNs and "Cyclo"s.

[QUOTE=airsquirrels;423034]Is there a prime between 57M and 74M, absolutely.

A mersenne prime? That is a different question[/QUOTE]

[QUOTE=ATH;423033][url]http://www.forbes.com/sites/kevinknudson/2016/01/19/new-record-prime-number-discovered/[/url]



That is nice to know :smile: We are almost certain to find one between 57M and 74M.[/QUOTE]

As usual, the news sites don't really know what they're talking about, and don't clearly state what they're trying to state.

If my calculations are right, we can expect about 0.18 more Mersenne primes in the 57M < p < 75M range. So it's not almost certain that we'll find another Mersenne there, rather it's almost certain (to be precise: ~85% likely) that we'll find no more Mersennes in that range.

It is certain that [i]some[/i] prime exists within that range. In fact, [URL="https://en.wikipedia.org/wiki/Prime_gap"]we can[/URL] make the stronger statement that between any two Mersenne candidates (once you get past the first few, anyway), multiple primes always exist (not just incredibly likely, but proven to). I'm sure we'll find one of the primes between these two Mersennes eventually, but it could be years before that happens.

:party::party::party:

Congratulations to us all, but especially to George, who now goes down in history as having contributed the most new Mersenne primes percentage-wise: 15 new ones compared to the 34 that were known before GIMPS' first discovery represents about a 44.1% increase, compared to Raphael Robinson's 5 new Mersenne primes added to 12 already known in 1952, a 41.7% increase. (Third place goes to Cataldi, who by 1588 had added 2 new ones to the 5 already known, a 40% increase.) Of course, it took George 20 years and tens of thousands of contributors to find these 15 primes, but even the coordination of a project this size deserves a lot of respect!

Congratulations also to Aaron, who well-deserves being credited by name, and to Curtis Cooper, who just keeps on crunching!