Category: Matrix Algebra

I am working on a factorization of the Thirteenth Fermat number which is 2 ^ 8192 + 1 and is 2,467 decimal digits in length. I am using Pollard’s factoring with cubic integers on the number (2 ^ 2731) ^ 3 + 2. I am also utilizing a homegrown variant of the venerable Pollard and Brent rho method and Arjen K. Lenstra’s Free LIP Elliptic Curve Method. I can factor the seventh Fermat number 2 ^ 128 + 1 in five to thirty minutes using my C# code. The factoring with cubic integers code is in C and uses Free-LIP.

Fermat factoring status (prothsearch.com)

The following is a run of Lenstra’s ECM algorithm:

== Data Menu ==
1 Simple Number
2 Fibonacci Sequence Number
3 Lucas Sequence Number
4 Exit
Enter option (1 – 4): 1
Enter a number to be factored: 2^8192+1
Enter a random number generator seed: 1
== Factoring Menu ==
1 Lenstra’s ECM
2 Lenstra’s Pollard-Rho
3 Pollard’s Factoring with Cubic Integers
Option (1 – 3): 1

2710954639361 p # digits 13
3603109844542291969 p # digits 19
Runtime (s) = 17015.344000

I aborted the previous computation due to the fact I was curious about the number of prime factors that could be found on personal computer. I will try a lot more calculation time in a future run. My homegrown application is able to at least find the first factor of Fermat Number 13.