## Fermat Number

A Binomial Number of the form . The first few for , 1, 2, ... are 3, 5, 17, 257, 65537, 4294967297, ... (Sloane's A000215). The number of Digits for a Fermat number is

 (1)

Being a Fermat number is the Necessary (but not Sufficient) form a number
 (2)

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

 (3)

Therefore, for a Prime , must be a Power of 2.

Fermat conjectured in 1650 that every Fermat number is Prime and Eisenstein (1844) proposed as a problem 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

 (4)

is Composite (Ribenboim 1996, p. 88). Numbers of the form are called generalized Fermat numbers (Ribenboim 1996, pp. 359-360).

Fermat numbers satisfy the Recurrence Relation

 (5)

can be shown to be Prime iff it satisfies Pépin's Test

 (6)

Pépin's Theorem
 (7)

is also Necessary and Sufficient.

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

 (8)

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
 (9)

If
 (10)

is the factored part of (where is the cofactor to be tested for primality), compute
 (11) (12) (13)

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

 (14)

where the are distinct Fermat primes. 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 only known Fermat Primes are

and it seems unlikely that any more exist.

Factoring Fermat numbers is extremely difficult as a result of their large size. In fact, only to have been complete factored, as summarized in the following table. Written out explicitly, the complete factorizations are

Here, the final large Prime is not explicitly given since it can be computed by dividing by the other given factors.

 Digits Factors Digits Reference 5 10 2 3, 7 Euler 1732 6 20 2 6, 14 Landry 1880 7 39 2 7, 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

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). Young and Buell (1988) discovered that is Composite, and Crandall et al. (1995) that is Composite. A current list of the known factors of Fermat numbers is maintained by Keller, and reproduced in the form of a Mathematica notebook by Weisstein. In these tables, since all factors are of the form , the known factors are expressed in the concise form . The number of factors for Fermat numbers for , 1, 2, ... are 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 5, ....

See also Cullen Number, Pépin's Test, Pépin's Theorem, Pocklington's Theorem, Polygon, Proth's Theorem, Selfridge-Hurwitz Residue, Woodall Number

References

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© 1996-9 Eric W. Weisstein
1999-05-26