# Fermat prime

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## Contents

## English[edit]

### Etymology[edit]

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

### Noun[edit]

**Fermat prime** (*plural* **Fermat primes**)

- (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 2
^{2n}+1 for some*n*≥ 0);*equivalently*, a number that is one more than two raised to some power (is expressible as 2^{n}+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 can possibly be prime is when is a power of 2, and so all
**Fermat primes**must have the form for integers . The numbers 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.

- It is not hard to prove that the only way that the number can possibly be prime is when is a power of 2, and so all
**2004**, T. W. Müller,*12: Parity patterns in Hecke groups and*, Thomas Wolfgang Müller (editor),**Fermat primes***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.

- Rather surprisingly, it turns out that
**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:*F*_{0}= 3,*F*_{1}= 5,*F*_{2}= 17,*F*_{3}= 257, and*F*_{4}= 65537.

- The next 28 Fermat numbers,
*F*_{5}through*F*_{32}, are known to be composite.

- Believe it or not, so far only five

#### Usage notes[edit]

The equivalence of the two definitions follows from the fact that, as can be demonstrated, for a number of the form to be prime it is *necessary* (though not *sufficient*) that for some . (See **Fermat number** on Wikipedia.Wikipedia )