# factorization of 2 numbers

I tried to factor the numbers $1013^{13}+331\#$ and $98!+76!+54!+32!+1$ with GMP-ECM because the quadratic sieve takes very much time. Perhaps someone factors one of those numbers for me. The second number has a prime factor of 25 digits, namely $1005693665747024080598971$. For those who are not familiar with the number theory notations: $!$ is the factorial $n!=1\cdot 2\cdot 3\cdot\ldots\cdot n$ and $n\#$ is the product of primes up to $n$: $n\#=2\cdot 3\cdot 5\cdot 7\cdot\ldots\cdot n$, if $n$ is a prime. And please do not ask why I am interested in these factorization. It is just curiousity because in the net it is stated that such numbers are already quite easy to factor. Useful programs are GMP-ECM or msieve downloadable from the net.

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The first, as 1013^13 + 331, gives 336 * prime number. Prehaps the # stands for something different? – wendy.krieger May 27 '13 at 7:04
Wendy : Please read the whole text, I explained the '#'. – Peter May 27 '13 at 7:07
Peter, this is the third account you created: math.stackexchange.com/users/79589/peter , math.stackexchange.com/users/79596/peter and this one. Please ask a moderator to merge them! I flagged your account to draw a moderator's attention... – draks ... May 27 '13 at 7:22
@draks... Mods can't do that anymore. The user has to contact SE directly math.stackexchange.com/help/user-merge – Double AA May 27 '13 at 7:24
I assume (the other) Peter might be referring to the fact that 512-bit moduli are considered to be weak for crypto schemes like RSA due to being factorable withing reasonable time-frame. – Peter Košinár May 27 '13 at 7:36

The first one took about 60 hours of NFS-ing on a moderately slow machine with unoptimized factoring settings while the machine was being used for other purposes too:

1013^13 + 331# = 13660997293197200778560512676198053659658942317253 *
469117945090530664165653254310752426123381482366114189309329070591871468575459348391


The other one took about 64 hours; the smallest factor being discovered by ECM pretty quickly and the rest of the time spent on NFS-ing (again, unoptimized, running in background, ...):

98!+76!+54!+32!+1 = 1005693665747024080598971 *
79146647927616255967923163095656379252005739066059813 *
118432315205629597504117998137752645490949757279339848413269947393226465090487


Altogether, this supports the statement that numbers of this size are quite easy to factor with current hardware. The time could be cut down to about one day on a dedicated, reasonably modern workstation and further cuts would be possible with better tuning of the factoring settings.

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+1 respect..... – draks ... Jun 4 '13 at 19:47