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# Fermat number

Pierre de Fermat wrote to Marin Mersenne on December 25, 1640 that:If I can determine the basic reason whyThis is usually taken to be the conjecture that every number of the form is prime. So we call these the3, 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].

**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(This gives a simple proof that there are infinitely many primes.)_{0}F_{1}F_{2}^{.}...^{.}F_{n-1}+2 = F_{n}.

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, 2001. New York, NY, 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.