Prime Decomposition of 7th Fermat Number

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Theorem

The prime decomposition of the $7$th Fermat number is given by:

\(\displaystyle 2^{\paren {2^7} } + 1\) \(=\) \(\displaystyle 340 \, 282 \, 366 \, 920 \, 938 \, 463 \, 463 \, 374 \, 607 \, 431 \, 768 \, 211 \, 457\) Sequence of Fermat Numbers
\(\displaystyle \) \(=\) \(\displaystyle 59 \, 649 \, 589 \, 127 \, 497 \, 217 \times 5 \, 704 \, 689 \, 200 \, 685 \, 129 \, 054 \, 721\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {116 \, 503 \, 103 \, 764 \, 643 \times 2^9 + 1} \paren {11 \, 141 \, 971 \, 095 \, 088 \, 142 \, 685 \times 2^9 + 1}\)


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 $F_7$ and $F_8$ are not prime, but without having established what the prime factors are.

The actual factors were not themselves determined until the work of Michael A. Morrison and John David Brillhart in $1970$.


Sources