# Factorisation of F8

F8 = 2^{28} + 1 = 2^{256} + 1 is the eighth
Fermat number.
In 1970, Morrison and Brillhart
factored the seventh Fermat number
by the continued fraction method,
leaving F8 as as the first Fermat number
which was neither prime nor completely factored.
F8 was known to be composite (with 78 decimal digits), but no
factors were known.

On
9 July 1980, Richard Brent and
John Pollard
factored F8, finding

F8 = 1238926361552897 . p_{62}

where p_{62} is a 62-decimal digit prime number.
A modification of Pollard's
"rho" method was used to find the
factorisation in two hours on a Univac 1100/42 computer. The
computation required about 23 million multiplications modulo F8.

## Epigram

The epigram

* I am now entirely persuaded to employ the method, *

a handy trick, on gigantic composite numbers

may be helpful if you wish to remember the smaller factor of F8.

## The Larger Factor

The larger factor p_{62} of F8 was first proved prime
by Hugh Williams using the method of
Williams and Judd.
Later, a simpler proof was provided by Brent, using the factorisation

p_{62} - 1 = 31618624099079 . p_{43}

## References and Further Details

A preliminary announcement of the factorisation of F8
appeared in
* AMS Abstracts* 1 (1980), 565, 80T-A212.
Further details can be found in the paper:

Richard P. Brent and John M. Pollard,
Factorization of the Eighth Fermat Number,
* Mathematics of Computation* 36 (1981), 627-630.

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