# Factorization of a $116$-digit number

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 ?

• Sep 28, 2018 at 7:24
• 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 :-) Sep 28, 2018 at 10:02
• @didgogns I invite you both to join in the project of Enzo Creti and to run siqs for this number. Sep 28, 2018 at 10:53
• It's might be better to use CADO-NFS. At any rate you'd be able to solve this problem within a day. Sep 28, 2018 at 16:42

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