# A New Factor of F13

F13 = 2^{213} = 2^{8192} + 1 is the 13-th
Fermat number.
At 19 June 1995 three prime factors of F13 were known:

F13 = 2710954639361 .
2663848877152141313 .
3603109844542291969 . c_{2417}

The 13-digit factor was found by Hallyburton and Brillhart in 1974.

The 19-digit factors were found by Richard Crandall on Zilla net

(a network of NeXT workstations)
in January and May 1991.

The quotient c_{2417}
is a composite number with 2417 decimal digits.

On 19 June 1995 I found a fourth factor with 27 decimal digits:

p_{27} = 319546020820551643220672513

The quotient is c2391, a composite number with 2391 decimal digits.

## The Computation

The computation to find p_{27} used Lenstra's
elliptic curve method (ECM)

implemented on a 40 Mhz IBM 80386 PC
with a Dubner Cruncher board to speed up the
multiple-precision arithmetic. Overall, it took about
47 days to find the factor
(my current program would take about 30 days).

## The Cruncher versus Supercomputers

On large numbers like F13, the Cruncher is about twice as fast as the
Fujitsu VP100 which I used to factorise
F11 back in 1988:

F11 = 319489 . 974849 . 167988556341760475137 . 3560841906445833920513 . p_{564},

where p_{564} is a 564-digit prime number.

Since the Cruncher costs only a few thousand dollars, it is a very
cost-effective

"factoring engine".

For more information about the Cruncher, please contact
Harvey Dubner <Firstname@dubner.com>

## Factorisation of F10

Recently I factored F10
using an implementation of ECM on a Sun workstation.
It could have been done on a Cruncher, but
F10 is too small to make really efficient use of a Cruncher.
Richard Brent

22 June 1995