Fermat prime

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

Fermat prime (plural Fermat primes)

1. (number theory) A prime number which is one more than two raised to a power which is itself a power of two (i.e., is expressible in the form 2²ⁿ+1 for some n ≥ 0); equivalently, a number that is one more than two raised to some power (is expressible as 2ⁿ+1) and is prime.

   Carl Friedrich Gauss proved the constructibility of the regular 17-gon in 1796. Five years later, he developed the theory of Gaussian periods in his Disquisitiones Arithmeticae. This theory allowed him to formulate a sufficient condition for the constructibility of regular polygons: "A regular n-gon can be constructed with compass and straightedge if n is the product of a power of 2 and any number of distinct Fermat primes." Gauss stated without proof that this condition was also necessary, but never published his proof. A full proof of necessity was given by Pierre Wantzel in 1837. The result is known as the Gauss–Wantzel theorem.^WP

   • 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, 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.

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 ${\displaystyle k}$ . (See Fermat number on Wikipedia.Wikipedia )

• Fermat number

• Mersenne prime

• Fermat Number on Wolfram MathWorld