Prime Decomposition of 8th Fermat Number

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Theorem

The prime decomposition of the $8$th Fermat number is given by:

\(\displaystyle 2^{\paren {2^8} } + 1\) \(=\) \(\displaystyle 115 \, 792 \, 089 \, 237 \, 316 \, 195 \, 423 \, 570 \, 985 \, 008 \, 687 \, 907 \, 853 \, 269 \, 984 \, 665 \, 640 \, 564 \, 039 \, 457 \, 584 \, 007 \, 913 \, 129 \, 639 \, 937\)
\(\displaystyle \) \(=\) \(\displaystyle 1 \, 238 \, 926 \, 361 \, 552 \, 897\)
\(\displaystyle \) \(\) \(\, \displaystyle \times \, \) \(\displaystyle 93 \, 461 \, 639 \, 715 \, 357 \, 977 \, 769 \, 163 \, 558 \, 199 \, 606 \, 896 \, 584 \, 051 \, 237 \, 541 \, 638 \, 188 \, 580 \, 280 \, 321\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {1 \, 209 \, 889 \, 024 \, 954 \times 2^{10} + 1}\)
\(\displaystyle \) \(\) \(\, \displaystyle \times \, \) \(\displaystyle \left({91 \, 271 \, 132 \, 534 \, 529 \, 275 \, 165 \, 198 \, 787 \, 304 \, 303 \, 609 \, 945 \, 362 \, 536 \, 661 \, 756 \, 043 \, 535 \, 430 \times 2^{10} + 1}\right)\)


Proof


Also see


Historical Note

In $1909$, James C. 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