## Factorization of the tenth Fermat number

161. R. P. Brent,
Factorization of the tenth Fermat number,
* Mathematics of Computation* 68 (1999), 429-451.
MR 99e:11154.
Preliminary (slightly different) version available as
* Factorization of the tenth and eleventh Fermat numbers*,
Technical Report TR-CS-96-02, CSL, ANU, Feb. 1996, 25 pp.

Abstract:
dvi (3K),
pdf (82K).

Paper:
dvi (53K),
pdf (363K),
ps (116K).

Technical Report:
dvi (60K),
pdf (539K),
ps (190K).

Transparencies:
dvi (12K),
pdf (218K),
ps (56K).

## Abstract for Paper

We describe the complete factorization of the tenth
Fermat number F_{10}
by the elliptic curve method (ECM).
The tenth Fermat number is a product
of four prime factors with 8, 10, 40 and 252 decimal digits.
8, 10, 40 and 252 decimal digits. The
40-digit factor was found after
about 140 Mflop-years of computation.
We also discuss the factorization of other Fermat numbers
by ECM, and summarize the complete factorizations of
F_{5} , ... , F_{11}.
## Abstract for Technical Report

We describe the complete factorization of the tenth and eleventh
Fermat numbers.
The tenth Fermat number is a product
of four prime factors with 8, 10, 40 and 252 decimal digits.
The eleventh Fermat number is a product of five prime factors
with 6, 6, 21, 22 and 564 decimal digits.
We also note a new
27-digit factor
of the thirteenth
Fermat number. This number has four known prime factors and a 2391-decimal
digit composite factor. All the new factors reported here were found by the
elliptic curve method (ECM).
The 40-digit factor
of the tenth Fermat number
was found after about 140 Mflop-years of computation. We discuss aspects of
the practical implementation of ECM, including the use of special-purpose
hardware, and note several other large factors found recently by ECM.
## Comments

A slight variation of the Fortran program used to factor F10 is available
here.
Related papers include:

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