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Fermat Number

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There are two definitions of the Fermat number. The less common is a number of the form 2^n+1 obtained by setting x=1 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 F_n=2^(2^n)+1. The first few for n=0, 1, 2, ... are 3, 5, 17, 257, 65537, 4294967297, ... (OEIS A000215). The number of digits for a Fermat number is

D(n)=|_[log(2^(2^n)+1)]+1_|
(1)
approx |_log(2^(2^n))+1_|
(2)
=1+|_2^nlog2_|.
(3)

For n=0, 1, ..., the numbers of digits in F_n are therefore 1, 1, 2, 3, 5, 10, 20, 39, 78, 155, 309, 617, 1234, ... (OEIS A057755). The numbers of digits in F_(10^n) for n=0, 1, ... are 1, 309, 381600854690147056244358827361, ... (OEIS A114484).

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

N_n=2^n+1
(4)

must have in order to be prime. This can be seen by noting that if N_n=2^n+1 is to be prime, then n cannot have any odd factors b or else N_n would be a factorable number of the form

2^n+1=(2^a)^b+1=(2^a+1)[2^(a(b-1))-2^(a(b-2))+2^(a(b-3))-...+1].
(5)

Therefore, for a prime N_n, n 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 F_n are known for n>=5. An anonymous writer proposed that numbers of the form 2^2+1, 2^(2^2)+1, 2^(2^(2^2))+1 were prime. However, this conjecture was refuted when Selfridge (1953) showed that

F_(16)=2^(2^(16))+1=2^(2^(2^(2^2)))+1
(6)

is composite (Ribenboim 1996, p. 88).

The only known Fermat primes are

F_0=3
(7)
F_1=5
(8)
F_2=17
(9)
F_3=257
(10)
F_4=65537
(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, as of 2022, only F_5 to F_(11) have been completely factored. The number of factors for Fermat numbers F_n for n=0, 1, 2, ... are 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 5, ... (OEIS A046052). Written out explicitly, the complete factorizations are

F_5=641·6700417
(12)
F_6=274177·67280421310721
(13)
F_7=59649589127497217·5704689200685129054721
(14)
F_8=1238926361552897·93461639715357977769163558199606896584051237541638188580280321
(15)
F_9=2424833·7455602825647884208337395736200454918783366342657·P99
(16)
F_(10)=45592577·6487031809·4659775785220018543264560743076778192897·P252
(17)
F_(11)=319489·974849·167988556341760475137·3560841906445833920513·P564
(18)

(OEIS A050922). Here, the final large prime is not explicitly given since it can be computed by dividing F_n 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 k2^n+1, the known factors are expressed in the concise form (k,n).

F_ndigitsfactorsdigitsreference
51023, 7Euler 1732
62026, 14Landry 1880
739217, 22Morrison and Brillhart 1975
878216, 62Brent and Pollard 1981
915537, 49, 99Manasse and Lenstra (In Cipra 1993)
1030948, 10, 40, 252Brent 1995
1161756, 6, 21, 22, 564Brent 1988

As of 2022, F_(12) has 6 known factors with C1133 remaining (where Cn denotes a composite number with n digits), F_(13) has 4 known factors with C2391 remaining, and F_(14) has one known factor with C4880 remaining (Keller).

By the early 1980s, F_n was known to be composite for all 5<=n<=32 with the exceptions n=20, 22, 24, 28, and 31 (Riesel 1994, Crandall et al. 2003). Young and Buell (1988) discovered that F_(20) is composite, Crandall et al. (1995) that F_(22) is composite, and Crandall et al. (2003) that F_(24) is composite (Crandall 1999; Borwein and Bailey 2003, pp. 7-8; Crandall et al. 2003). In 1997, Taura found a small factor of F_(28) (Crandall et al. 2003, Keller), and a small factor of F_(31) was also found. As of 2022, it is known that F_n is composite for all 5<=n<=33 (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: F_(20) and F_(24) (Keller; cf. Crandall et al. 2003).

Ribenboim (1996, pp. 89 and 359-360) defines generalized Fermat numbers as numbers of the form a^(2^n)+1 with a>2 even, while Riesel (1994, pp. 102 and 415) defines them more generally as numbers of the form a^(2^n)+b^(2^n).

Fermat numbers satisfy the recurrence relation

F_m=F_0F_1...F_(m-1)+2.
(19)

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

3^(2^(2^n-1))=-1 (mod F_n)
(20)

is also both necessary and sufficient.

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

2^(n+1)K+1,
(21)

where K 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

2^(n+2)L+1
(22)

for n>1. Factors of Fermat numbers are therefore Proth primes since they are of the form k·2^n+1, as long as they also satisfy the additional condition k odd and 2^n>k.

If

F=p_1p_2...p_r
(23)

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

A=3^(F_n-1) (mod F_n)
(24)
B=3^(F-1) (mod F_n)
(25)
R=A-B (mod C).
(26)

Then if R=0, the cofactor is a probable prime to the base 3^F; otherwise C 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 N given by

N=2^kF_0...F_n,
(27)

where k is nonnegative and the F_i are zero or more distinct Fermat primes (as stated by Gauss and first published by Wantzel 1836). This is equivalent to the statement that the trigonometric functions sin(kpi/N), cos(kpi/N), etc., can be computed in terms of finite numbers of additions, multiplications, and square root extractions iff N is of the above form.

The last d digits of {F_k,F_(k+1),...} (where k is the smallest integer such that F_k has d digits) are eventually periodic for d=1, 2, ... with periods 1, 4, 20, 100, 500, 2500, ... (OEIS A005054; Koshy 2002-2003).

See also

Cullen Number, Fermat Polynomial, Fermat Prime, Generalized Fermat Number, Near-Square Prime, Pépin's Test, Pépin's Theorem, Pocklington's Theorem, Polygon, Proth Prime, Proth's Theorem, Selfridge-Hurwitz Residue, Woodall Number

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References

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Fermat Number

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

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