# Fermat number

Pierre de Fermat wrote to Marin Mersenne on December 25, 1640 that:

If I can determine the basic reason why3, 5, 17, 257, 65 537, ...,are prime numbers, I feel that I would find very interesting results, for I have already found marvelous things [along these lines] which I will tell you about later. [Archibald1914].

This is usually taken to be the conjecture that every number of the form is prime. So we call these the **Fermat numbers**, and when a number of this form is prime, we call it a **Fermat prime**.

The only known Fermat primes are the first five Fermat numbers: F_{0}=3, F_{1}=5, F_{2}=17, F_{3}=257, and F_{4}=65537. A simple heuristic shows
that it is likely that these are the only Fermat primes (though many folks like Eisenstein thought otherwise).

In 1732 Euler discovered 641 divides F_{5}. It only takes two trial divisions to find this factor because Euler showed that every divisor of a Fermat number F_{n} with *n* greater than 2 has the form *k*^{.}2^{n+1}+1 (exponent improved to *n*+2 by Lucas). In the case of F_{5} this is 128*k*+1, so we would try 257 and 641 (129, 385, and 513 are not prime). Now we know that all of the Fermat numbers are composite for the other *n* less than 31. The quickest way to check a Fermat number for primality (if trial division fails to find a small factor) is to use Pepin's test.

Gauss proved that a regular polygon of *n* sides
can be inscribed in a circle with Euclidean methods (e.g.,
by straightedge and compass) if and only if *n* is a
power of two times a product of distinct Fermat primes.

The Fermat numbers are pairwise relatively prime, as can be seen in the following identity:

F_{0}F_{1}F_{2}^{.}...^{.}F_{n-1}+2 = F_{n}.

(This gives a simple proof that there are infinitely many primes.)

The quickest way to find out if a Fermat number is prime, is to use Pepin's test [Pepin77]. It is not yet known if there are infinitely many Fermat primes, but it seems likely that there are not.

The book [KLS2001] is an excellent source of information on these numbers.

**See Also:** PepinsTest, Heuristic, PierpontPrime

**References:**

- Archibald1914
R. C. Archibald, "Remarks on Klien's `Famous problems of elementary geometry',"Amer. Math. Monthly,21(1914) 247--259.- Carmichael19
R. D. Carmichael, "Fermat numbersF_{n}=2^{2n}+1,"Amer. Math. Monthly,26(1919) 137--146.- CDNY95
R. Crandall,J. Doenias,C. NorrieandJ. Young, "The twenty-second Fermat number is composite,"Math. Comp.,64(1995) 863--868.MR 95f:11104- CMP2003
R. E. Crandall,E. W. MayerandJ. S. Papadopoulos, "The twenty-fourth Fermat number is composite,"Math. Comp.,72(2003) 1555--1572. (Abstract available)- Gostin95
G. B. Gostin, "New factors of Fermat numbers,"Math. Comp.,64(1995) 393--395.MR 95c:11151- HB1975
J. C. Hallyburton, Jr.andJ. Brillhart, "Two new factors of Fermat numbers,"Math. Comp.,29(1975) 109--112. Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday.MR 51:5460- HS61
A. HurwitzandJ. L. Selfridge, "Fermat numbers and perfect numbers,"Notices Amer. Math. Soc.,8(1961) 601. Abstract 587-104.- Keller83
W. Keller, "Factors of Fermat numbers and large primes of the formk· 2^{n}+1,"Math. Comp.,41(1983) 661-673.MR 85b:11117- Keller92
W. Keller, "Factors of Fermat numbers and large primes of the formk· 2^{n}+1 ii," Hamburg, (September 1992) Manuscript.- KLS2001
M. Krízek,F. LucaandL. Somer,17 lectures on Fermat numbers: from number theory to geometry, CMS Books in Mathematics Vol, 9, Springer-Verlag, New York, NY, 2001. pp. xvii + 257, ISBN 0-387-95332-9.MR 2002i:11001- LLMP93
A. K. Lenstra,Lenstra, Jr., H. W.,M. S. ManasseandJ. M. Pollard, "The factorization of the ninth Fermat number,"Math. Comp.,61(1993) 319-349. Addendum,Math. Comp.64(1995), 1357.MR 1 303 085- Pepin77
T. Pepin, "Sur la formule 2^{2n}+1,"C. R. Acad. Sci. Paris,85(1877) 329-331.- Robinson54
R. M. Robinson, "Mersenne and Fermat numbers,"Proc. Amer. Math. Soc.,5(1954) 842-846. [Announces the discovery of the 13th through 17th Mersenne primes--the first Mersenne primes found by electronic computer.]- Robinson57
R. M. Robinson, "Factors of Fermat numbers,"Math. Tables Aids Comput.,11(1957) 21-22.- Robinson58
R. M. Robinson, "A report on primes of the formk· 2^{n}+1 and on factors of Fermat numbers,"Proc. Amer. Math. Soc.,9(1958) 673--681.MR 20:3097- Selfridge53
J. L. Selfridge, "Factors of Fermat numbers,"Math. Tables Aids Comput.,7(1953) 274-275.- Shippee1978
D. E. Shippee, "Four new factors of Fermat numbers,"Math. Comp.,32:143 (1978) 941. (Abstract available)- SS1981
Shorey, T. N.andStewart, C. L., "On divisors of Fermat, Fibonacci, Lucas and Lehmer numbers. II,"J. London Math. Soc. (2),23:1 (1981) 17--23.MR 82m:10025- Stewart1977
C. L. Stewart, "On divisors of Fermat, Fibonacci, Lucas and Lehmer numbers,"Proc. Lond. Math. Soc.,35:3 (1977) 425--447.MR 58:10694- Stewart1983
Stewart, C. L., "On divisors of Fermat, Fibonacci, Lucas and Lehmer numbers. III,"J. London Math. Soc. (2),28:2 (1983) 211--217.MR 85g:11021- Stewart1983
Stewart, C. L., "On divisors of Fermat, Fibonacci, Lucas and Lehmer numbers. III,"J. London Math. Soc. (2),28:2 (1983) 211--217.MR 85g:11021- Wrathall64
C. P. Wrathall, "New factors of Fermat numbers,"Math. Comp.,18(1964) 324--325.MR 29:1167- YB88
J. YoungandD. A. Buell, "The twentieth Fermat number is composite,"Math. Comp.,50(1988) 261--263.MR 89b:11012- Young98
J. Young, "Large primes and Fermat factors,"Math. Comp.,67:244 (1998) 1735--1738.MR 99a:11010Abstract:A systematic search for large primes has yielded the largest Fermat factors known.