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What is the prime factorization of this number : $$2510840694154672305534315769283066566440942177785613805158$$ $$3255420347077336152767157884641533283220471088892806902579$$ ?

If we concatenate the Mersenne-numbers $\ \ M(193)=2^{193}-1\ \ $ and $\ \ M(192)=2^{192}-1\ \ $ and divide by $5$, we get this $116$-digit composite number.

I tried to factor it with the $1.34$-version from yafu, but ecm did not give a result. The quadratic sieve is very time-consuming.

The smallest prime factor of this number probably has more than $40$ digits.

How can I factor this number ?

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    $\begingroup$ Related : math.stackexchange.com/questions/2635516/… $\endgroup$
    – Peter
    Sep 28, 2018 at 7:24
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    $\begingroup$ It's 116 digits composite with ECM done, then I think GNFS will be much faster than Quadratic sieve. Download a copy of GGNFS, edit yafu.ini to indicate the location of GGNFS, run "tune()" command then you will be able to run NFS on your composite. Also, "-threads" option helps :-) $\endgroup$
    – didgogns
    Sep 28, 2018 at 10:02
  • $\begingroup$ @didgogns I invite you both to join in the project of Enzo Creti and to run siqs for this number. $\endgroup$
    – Peter
    Sep 28, 2018 at 10:53
  • $\begingroup$ It's might be better to use CADO-NFS. At any rate you'd be able to solve this problem within a day. $\endgroup$
    – Yong Hao Ng
    Sep 28, 2018 at 16:42

1 Answer 1

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I factored this 116-digit composite, the factorization is $$C116=179870387995451933124857321125705471410636663412965107437\times139591664983685893216072195834659800165004530288592786103967$$

On my laptop with i3-5005U using yafu 1.34 and GGNFS, the factorization took about 14 hours.

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    $\begingroup$ Thank you for your calculation ! (+1) $\endgroup$
    – Peter
    Sep 29, 2018 at 9:57
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    $\begingroup$ Thank you for the achievement! $\endgroup$
    – Enzo Creti
    Sep 29, 2018 at 15:39

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