Section: Fermat Numbers

The Fermat numbers are defined by \[F_n:=2^{2^n}+1,\hspace{4em}(\text{for }n\ge 0)\]

The sequence of Fermat numbers begins with \(3\), \(5\), \(17\), \(257\), \(65537\), \(4294967297\), ...

Historical Notes

1640, Fermat wrote a letter to Frénicle de Bessy and stated in it a conjecture that \(F_n\) is prime for each \(n\), which is true for \(n\le 4\). 1732, Euler showed that \(F_5=4294967297\) is not prime since \(641\mid F_5\).

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