A New Factor of F13

F13 = 2213 = 28192 + 1 is the 13-th Fermat number.

At 19 June 1995 three prime factors of F13 were known:

F13 = 2710954639361 . 2663848877152141313 . 3603109844542291969 . c2417

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 c2417 is a composite number with 2417 decimal digits.

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

p27 = 319546020820551643220672513

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

The Computation

The computation to find p27 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 . p564,

where p564 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