Unforgettable Fermat Factors
In the March, 1998 issue of the Mathematical Gazette, John
Pollard gives several mnemonics for memorizing factors of Fermat
numbers in his paper, Unforgettable Fermat Factors.
The number of letters in each word of the mnemonics correspond
to the digits of Fermat factors. A few of the mnemonics are below 
to see the others, get a copy of the paper (highly recommended)!
The F4 mnemonic is due to Richart Brent.
 "Fermat prime, maybe the largest?"
corresponding to
the largest known Fermat prime, F4 = 65537.

"I am now entirely persuaded to employ rho method,
a handy trick, on gigantic composite numbers"
corresponding to the prime factor 1238926361552897 of F8.

"MASSIVE TEAM BROKE NINTH FERMAT!
It factored as three primes, June fifteen (forenoon)
nineteen nine oh. Actually one can explain the
algorithm quite quickly and easily, er . . Well,
space here precludes a detailed account  candidly,
the big double search was done by Number Field
Sieving!"
corresponding to the prime factor
7455602825647884208337395736200454918783366342657
of F9. [periods (full stops) and exclamation
marks denote single zeros, except at the end. Two dots denote
double zero. Other punctuation is ignored].