Prime Decomposition of 7th Fermat Number
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Theorem
The prime decomposition of the $7$th Fermat number is given by:
| \(\ds 2^{\paren {2^7} } + 1\) | \(=\) | \(\ds 340 \, 282 \, 366 \, 920 \, 938 \, 463 \, 463 \, 374 \, 607 \, 431 \, 768 \, 211 \, 457\) | Sequence of Fermat Numbers | |||||||||||
| \(\ds \) | \(=\) | \(\ds 59 \, 649 \, 589 \, 127 \, 497 \, 217 \times 5 \, 704 \, 689 \, 200 \, 685 \, 129 \, 054 \, 721\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {116 \, 503 \, 103 \, 764 \, 643 \times 2^9 + 1} \paren {3^5 \times 5 \times 12497 \times 733803 839347 \times 2^9 + 1}\) |
Also see
Historical Note
In $1909$, James Caddall 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
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $257$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $257$