Fermat Number

The term "Fermat number" has two definitions. The less common is a number of the form obtained by setting in a Fermat-Lucas polynomial for positive integers , the first few of which are 3, 5, 9, 17, 33, ... (OEIS A000051).

The much more commonly encountered Fermat numbers are a special case, given by the binomial number of the form . The first few for , 1, 2, ... are 3, 5, 17, 257, 65537, 4294967297, ... (OEIS A000215). The number of digits for a Fermat number is

			(1)
			(2)
			(3)

where denotes the base-10 logarithm and is the floor function. For , 1, ..., the numbers of digits in are therefore 1, 1, 2, 3, 5, 10, 20, 39, 78, 155, 309, 617, 1234, ... (OEIS A057755). The numbers of digits in for , 1, ... are 1, 309, 381600854690147056244358827361, ... (OEIS A114484).

Being a Fermat number is the necessary (but not sufficient) form a number

(4)

with must have in order to be prime. This can be seen by noting that if is to be prime, then cannot have any odd factors or else would be a factorable number of the form

(5)

Therefore, for a prime , must be a power of 2. No two Fermat numbers have a common divisor greater than 1 (Hardy and Wright 1979, p. 14). Their reciprocal sum is the reciprocal Fermat constant.

Fermat conjectured in 1650 that every Fermat number is prime and Eisenstein proposed as a problem in 1844 the proof that there are an infinite number of Fermat primes (Ribenboim 1996, p. 88). At present, however, only composite Fermat numbers are known for . An anonymous writer proposed that numbers of the form , , were prime. However, this conjecture was refuted when Selfridge (1953) showed that

(6)

is composite (Ribenboim 1996, p. 88).

The only known Fermat primes are

			(7)
			(8)
			(9)
			(10)
			(11)

(OEIS A019434).

Factoring Fermat numbers is extremely difficult as a result of their large size. In fact, as of 2026, only to have been completely factored. The number of factors for Fermat numbers for , 1, 2, ... are 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 5, ... (OEIS A046052). Written out explicitly, the complete prime factorizations are

			(12)
			(13)
			(14)
			(15)
			(16)
			(17)
			(18)

(OEIS A050922). Here, the final large prime is not explicitly given since it can be computed by dividing by the other given factors. The following table summarizes the properties of these completely factored Fermat numbers.

	digits	factors	digits	reference
5	10	2	3, 7	Euler 1732
6	20	2	6, 14	Landry 1880
7	39	2	17, 22	Morrison and Brillhart 1975
8	78	2	16, 62	Brent and Pollard 1981
9	155	3	7, 49, 99	Manasse and Lenstra (In Cipra 1993)
10	309	4	8, 10, 40, 252	Brent 1995
11	617	5	6, 6, 21, 22, 564	Brent 1988

For , 2, ..., the smallest factors of the Fermat numbers are 5, 17, 257, 65537, 641, 274177, 59649589127497217, 1238926361552897, 2424833, ... (OEIS A093179), while the largest are 5, 17, 257, 65537, 6700417, 67280421310721, 5704689200685129054721, ... (OEIS A070592).

Other tables of known factors of Fermat numbers are given by Keller (1983), Brillhart et al. (1988), Young and Buell (1988), Riesel (1994), and Pomerance (1996). A current list of the known factors of Fermat numbers is maintained by Keller. In these tables, since all factors are of the form , the known factors are expressed in the concise form .

As of 2026, has 6 known factors with C1133 remaining (where C denotes a composite number with digits), has 4 known factors with C2391 remaining, has one known factor with C4880 remaining, and has three known factors with C9808 remaining (Keller).

By the early 1980s, was known to be composite for all with the exceptions , 22, 24, 28, and 31 (Riesel 1994, Crandall et al. 2003). Young and Buell (1988) discovered that is composite, Crandall et al. (1995) that is composite, and Crandall et al. (2003) that is composite (Crandall 1999; Borwein and Bailey 2003, pp. 7-8; Crandall et al. 2003). In 1997, Taura found a small factor of (Crandall et al. 2003, Keller), and a small factor of was also found. As of 2026, it is known that is composite for all (cf. Crandall et al. 2003).

There are currently two Fermat numbers that are known to be composite, but for which no single factor is known: and (Keller; cf. Crandall et al. 2003).

Ribenboim (1996, pp. 89 and 359-360) defines generalized Fermat numbers as numbers of the form with even, while Riesel (1994, pp. 102 and 415) defines them more generally as numbers of the form .

Fermat numbers satisfy the recurrence relation

(19)

For , can be shown to be prime iff it satisfies Pépin's test. Pépin's theorem

(20)

is also both necessary and sufficient.

In 1770, Euler showed that any prime factor of for must have the form

(21)

where is a positive integer. In 1878, Lucas increased the exponent of 2 by one, showing that prime factors of Fermat numbers must be of the form

(22)

for . Prime factors of Fermat numbers are therefore Proth primes since they are of the form , as long as they also satisfy the additional condition odd and .

If

(23)

is the factored part of for (where is the cofactor to be tested for primality), compute

			(24)
			(25)
			(26)

Then if , the cofactor is a probable prime to the base ; otherwise is composite.

A regular polygon with sides is a constructible polygon iff

(27)

where is a nonnegative integer and the are zero or more distinct Fermat primes, as stated by Gauss and first published by Wantzel (1837). This is equivalent to the statement that the trigonometric functions and can be computed in terms of finite numbers of additions, multiplications, and square root extractions iff is of the above form.

The last digits of (where is the smallest integer such that has at least digits) are eventually periodic for , 2, ... with periods 1, 4, 20, 100, 500, 2500, ... (OEIS A005054; Koshy 2002-2003).