Fermat number

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Fermat number

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Named after French lawyer and amateur mathematician Pierre de Fermat (1601–1665).

Fermat number (plural Fermat numbers)

1. (number theory) An integer which is one more than two raised to a power which is itself a power of two (i.e., is expressible in the form ${\displaystyle 2^{2^{n}}+1}$ for some ${\displaystyle n\geq 0}$); equivalently, a number that is one more than two raised to some power (is expressible as ${\displaystyle 2^{n}+1}$) and is prime.

   Hyponym: Fermat prime

The equivalence of the two definitions follows from the fact that, as can be demonstrated, for a number of the form ${\displaystyle 2^{n}+1}$ to be prime it is necessary (though not sufficient) that ${\displaystyle n=2^{k}}$ for some ${\displaystyle k}$.