Fermat prime

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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 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.
    • 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.

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 Wikipedia-logo-v2.svg Fermat number on Wikipedia.Wikipedia )

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