Fermat Number

There are two definitions of the Fermat number. The less common is a number of the form  obtained by setting  in a Fermat polynomial, 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)

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)

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).

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), and it seems unlikely that any more will be found using current computational methods and hardware.

Factoring Fermat numbers is extremely difficult as a result of their large size. In fact, 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 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 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).

The following table summarizes the properties of these completely factored Fermat numbers. 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 .

	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

 has 5 known factors with C1187 remaining (where C denotes a composite number with  digits).  has 4 known factors with C2391 remaining.  has no known factors but is composite.

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 factors of  was also found. It is therefore currently known that  is composite for all  (Crandall et al. 2003).

There are currently four Fermat numbers that are known to be composite, but for which no single factor is known: , , , and  (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)

 can be shown to be prime iff it satisfies Pépin's test

(20)

Pépin's theorem

(21)

is also necessary and sufficient.

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

(22)

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

(23)

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

(24)

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

			(25)
			(26)
			(27)

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

In order for a polygon to be circumscribed about a circle (i.e., a constructible polygon), it must have a number of sides  given by

(28)

where the  are distinct Fermat primes (as stated by Gauss and first published by Wantzel 1836). This is equivalent to the statement that the trigonometric functions , , etc., 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  digits) are eventually periodic for , 2, ... with periods 1, 4, 20, 100, 500, 2500, ... (OEIS A005054; Koshy 2002-2003).

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