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Fermat Prime

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A Fermat prime is a Fermat number F_n=2^(2^n)+1 that is prime. Fermat primes are therefore near-square primes.

Fermat conjectured in 1650 that every Fermat number is prime and Eisenstein in 1844 proposed as a problem the proof that there are an infinite number of Fermat primes (Ribenboim 1996, p. 88). At present, however, the only Fermat numbers F_n for n>=5 for which primality or compositeness has been established are all composite.

The only known Fermat primes are

F_0=3
(1)
F_1=5
(2)
F_2=17
(3)
F_3=257
(4)
F_4=65537
(5)

(OEIS A019434), and it seems unlikely that any more will be found using current computational methods and hardware. It follows that 2^n+1 is prime for the special case n=0 together with the Fermat prime indices, giving the sequence 2, 3, 5, 17, 257, and 65537 (OEIS A092506).

2^(2^n)+1 is a Fermat prime if and only if the period length of 1/(2^(2^n)+1) is equal to 2^(2^n). In other words, Fermat primes are full reptend primes.

See also

Constructible Polygon, Fermat Number, Full Reptend Prime, Generalized Fermat Number, Integer Sequence Primes, Mersenne Prime, Near-Square Prime, Pierpont Prime, Sierpiński Sieve, Trigonometry Angles

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References

Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, 1996.Robinson, R. M. "Mersenne and Fermat Numbers." Proc. Amer. Math. Soc. 5, 842-846, 1954.Sloane, N. J. A. Sequences A019434 and A092506 in "The On-Line Encyclopedia of Integer Sequences."

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Fermat Prime

Cite this as:

Weisstein, Eric W. "Fermat Prime." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FermatPrime.html

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