Prime Decomposition of 8th Fermat Number/Historical Note

Historical Note on Prime Decomposition of 8th Fermat Number

In $1909$, James Caddall Morehead and Alfred E. Western reported in Bulletin of the American Mathematical Society that they had proved that $F_8$ is not prime, but without having established what the prime factors are.

This factorisation was accomplished by John Michael Pollard and Richard Peirce Brent in $1981$.

Using a Monte Carlo method, they determined the prime factors, but were unable at the time to demonstrate that the larger factor was actually prime.

They devised a mnemonic for the smaller factor:

I am now entirely persuaded to employ the method, a handy trick, on gigantic composite numbers.

$40$ years later, a factorisation tool freely available online, running on a machine of modest specifications, can determine the primality of the larger factor (and indeed, $F_8$ itself) practically instantaneously.

Sources

• 1981: John M. Pollard and Richard P. Brent: Factorization of the Eighth Fermat Number (Math. Comp. Vol. 36: pp. 627 – 630)  www.jstor.org/stable/2007666

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $257$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $257$