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 F10
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
F5 , ... , F11.
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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