Fermat prime

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Fermat number

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

Fermat prime (plural Fermat primes)

1. (number theory) A Fermat number that is prime.

   Coordinate terms: Mersenne prime, Sophie Germain prime

   • 2001, I. Martin Isaacs, Geometry for College Students, American Mathematical Society, page 201:

     It is not hard to prove that the only way that the number ${\displaystyle 1+2^{e}}$ can possibly be prime is when ${\displaystyle e}$ is a power of 2, and so all Fermat primes must have the form ${\displaystyle 1+2^{2^{a}}}$ for integers ${\displaystyle a\geq 0}$. The numbers ${\displaystyle F_{a}=1+2^{2^{a}}}$ are called Fermat numbers, and although it is true that every Fermat prime is a Fermat number, it is certainly not true that every Fermat number is prime.

   • 2004, T. W. Müller, “12: Parity patterns in Hecke groups and Fermat primes”, in Thomas Wolfgang Müller, editor, Groups: Topological, Combinatorial and Arithmetic Aspects, Cambridge University Press, page 327:

     Rather surprisingly, it turns out that Fermat primes play an important special role in this context, a phenomenon hitherto unobserved in the arithmetic theory of Hecke groups, and, as a byproduct of our investigation, several new characterizations of Fermat primes are obtained.

   • 2013, Dean Hathout, Wearing Gauss’s Jersey, Taylor & Francis (CRC Press), page xi,

     Believe it or not, so far only five Fermat primes are known:

     F₀ = 3, F₁ = 5, F₂ = 17, F₃ = 257, and F₄ = 65537.

     The next 28 Fermat numbers, F₅ through F₃₂, are known to be composite.

• Fermat number

• Fermat Number on Wolfram MathWorld