[QUOTE=TheMawn;389828]I'm a bit unfamiliar with Probable Primes. How Probable is Probable? Is there a [I]realistic[/I] chance that any of the PRP's for the "fully factored" Mersennes are actually composite?[/QUOTE]
Actually the chances are not so remote. It is true that for a number of that size to be strong pseudoprime to some bases,[URL="http://primes.utm.edu/notes/prp_prob.html"] the chances are extraordinarily low,[/URL] even lower than Retina said, but Mersenne factors don't behave like general numbers of their sizes. It is quite common for a composite mersenne factor to be 2-PSP, for example all 3 composite combinations of 233, 1103 and 2089 (prime factors of M29) are pseudoprimes base 2. This seems to happen more often for p=1 (mod 4), but it happens often enough for p=3 (mod 4), for example 514447 is a 2-PSP (is the product of 359 and 1433, prime factors of M179). Interesting enough, you can find small multiple-base-PSP very easy for small composite factors of mersenne numbers, for example 10974881 is both 2-PRP and 3-PRP, but it not a prime (it is a PSP product of 1913 and 5737, prime factors of M239).

And so on. There can be easily found small mersenne factors which are PRP to many bases, but still composite. So, there IS a remote chance that those big PRPs are still PSP. I guess we will not know very soon.

Anyhow, the result is nice, and worth of congratulations.

[QUOTE=LaurV;389837]Actually the chances are not so remote. It is true that for a number of that size to be strong pseudoprime to some bases,[URL="http://primes.utm.edu/notes/prp_prob.html"] the chances are extraordinarily low,[/URL] even lower than Retina said, but Mersenne factors don't behave like general numbers of their sizes. It is quite common for a composite mersenne factor to be 2-PSP, for example all 3 composite combinations of 233, 1103 and 2089 (prime factors of M29) are pseudoprimes base 2. This seems to happen more often for p=1 (mod 4), but it happens often enough for p=3 (mod 4), for example 514447 is a 2-PSP (is the product of 359 and 1433, prime factors of M179). Interesting enough, you can find small multiple-base-PSP very easy for small composite factors of mersenne numbers, for example 10974881 is both 2-PRP and 3-PRP, but it not a prime (it is a PSP product of 1813 and 5737, prime factors of M239). [/QUOTE]

Base 2 should not be used for Mersenne cofactors. They [B]all[/B] pass it.

Let p be a prime and f = 2[SUP]s[/SUP]kp+1 be a composite factor of 2^p-1 (where k is odd).

We know that 2^p == 1 (mod f). Raising both sides by k, we get

2^(pk) == 1 (mod f)
i.e. 2^((f-1)/2[SUP]s[/SUP]) == 1 (mod f)

IOW, f passes Base-2 strong PRP test.

What PRP tests have been done on M488,441?

Chris

The default from Prime95. There is a setting in the program to change the base. If someone wants to do it, go ahead.

[CODE] ./pfgw64 -tc -q"(2^488441-1)/(61543567*30051203516986199)"
PFGW Version 3.7.7.64BIT.20130722.x86_Dev [GWNUM 27.11]

Primality testing (2^488441-1)/(61543567*30051203516986199) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Running N+1 test using discriminant 13, base 1+sqrt(13)
(2^488441-1)/(61543567*30051203516986199) is Fermat and Lucas PRP! (2068.5123s+0.0169s)
[/CODE]

I ran my own algorithm written in GMP:

[CODE]Likely prime with a=0

real 86m52.781s
user 84m37.781s
sys 1m32.782s
[/CODE]

Which means jacobiSymbol(-1,n)==-1 and (x+2)^(n+1)==5 (mod n, x^2+1).

This implies a 5-PRP test and the test: x^(n+1)==1 (mod n, x^2-(6/5)*x+1) :smile:

Another big cofactor of Mersenne number which is probable prime:

(2^440399-1)/(16210820281161978209*31518475633*880799)

Another fully factored small Mersenne:[URL="http://www.mersenne.ca/exponent.php?exponentdetails=10433"] M10433[/URL] (and [URL="http://factordb.com/index.php?query=M10433"]at FactorDB[/URL])
M10433 = 146063 · 7345550506166399 · 17578384916225511229570561 · [COLOR=Blue]407523153578238773059225963827711400649[/COLOR][SUB]<39>[/SUB] · [COLOR=Blue]P3056[/COLOR]

One more:
[URL="http://factordb.com/index.php?query=2%5E35339-1"]M35339[/URL] = 5776625742089 · 291148630508887 · 7028028455954046211351 · [COLOR="Blue"]4153830438466899077960892137[SUB]<28>[/SUB] · P10562[/COLOR]

[QUOTE=Batalov;400573]One more:
[URL="http://factordb.com/index.php?query=2%5E35339-1"]M35339[/URL] = 5776625742089 · 291148630508887 · 7028028455954046211351 · [COLOR="Blue"]4153830438466899077960892137[SUB]<28>[/SUB] · P10562[/COLOR][/QUOTE]
Excellent findings!!! It is not clear to me why the report at [url]http://www.mersenne.org/report_exponent/?exp_lo=35339&exp_hi=&full=1[/url] does not show the finder of the 28-digit prime factor.

I tried different options for submitting a factor, and all of them don't list the finder (me :-)
I am now simply submitting a single line
[QUOTE]M35339 has a factor: 4153830438466899077960892137[/QUOTE]
into manual results form - either on mersenne.ca or mersenne.org (while logged in!).

I tried pasting the whole ECM output - the result is the same (or worse - not parsed out).

It doesn't matter to me. It is a valid contender for the [URL="http://primes.utm.edu/top20/page.php?id=49"]Mersenne cofactor[/URL] top20; this does matter. The cofactor is proven before submitting, of course.