# Factorisation of F10

F10 = 2^{210} = 2^{1024} + 1 is the tenth
Fermat number.
At 20 October 1995 two small prime factors of F10 were known:

F10 = 45592577 . 6487031809 . c_{291}

Here c_{291} is a 291-decimal digit composite number.
The small factors were found by Selfridge (1953) and Brillhart (1962).
The composite number c_{291} has been the "most wanted number" ever
since the factorization of F9 in 1990.

On 20 October I found a 40-digit prime factor p_{40}
of c_{291}:

p_{40} = 4659775785220018543264560743076778192897

The quotient is a prime p_{252} of 252 decimal digits.
Thus we have the complete factorization

F10 = 45592577 . 6487031809
. 4659775785220018543264560743076778192897
. p_{252}

## The Computation

The computation to find p_{40} used Lenstra's
elliptic curve method (ECM)

It took about 3000 curves with first-phase limit 2000000
to find the factor.
This total includes some curves with smaller limits, appropriately weighted.
Each curve with limit 2000000 requires 46 million multiplications
modulo the 291-digit number to be factored,
and this takes about 11 hours on a Sun Viking Sparc.
Overall, the computation took about 140 Mflop-years.
## Further Details

See my paper
Factorization of the tenth Fermat number,
* Mathematics of Computation 68 (1999)*, 429-451.

## Factorisation of F9 by ECM

It is possible to find the factorisation of F9
by ECM, although it was first found by SNFS.
## Factor of F13

Using ECM on a Dubner Cruncher,
I recently found a 27-digit factor
of F13.
Richard Brent

26 October 1995

(revised 15 Feb 1996)