Fermat number
[5] The largest Fermat number known to be composite is F18233954, and its prime factor 7 × 218233956 + 1 was discovered in October 2020.
Boklan and Conway published a more precise analysis suggesting that the probability that there is another Fermat prime is less than one in a billion.
But Fermat numbers grow so rapidly that only a handful of them can be tested in a reasonable amount of time and space.
If N = Fn > 3, then the above Jacobi symbol is always equal to −1 for a = 3, and this special case of Proth's theorem is known as Pépin's test.
Because of Fermat numbers' size, it is difficult to factorize or even to check primality.
Pépin's test gives a necessary and sufficient condition for primality of Fermat numbers, and can be implemented by modern computers.
Distributed computing project Fermatsearch has found some factors of Fermat numbers.
Yves Gallot's proth.exe has been used to find factors of large Fermat numbers.
Édouard Lucas, improving Euler's above-mentioned result, proved in 1878 that every factor of the Fermat number
[11] In 1904, Cipolla showed that the product of at least two distinct prime or composite Fermat numbers
Let Gp denote the group of non-zero integers modulo p under multiplication, which has order p − 1.
Notice that 2 (strictly speaking, its image modulo p) has multiplicative order equal to
Hence (as was known to Carl Friedrich Gauss), 2 is a quadratic residue modulo p, that is, there is integer a such that
(Luca 2000) The series of reciprocals of all prime divisors of Fermat numbers is convergent.
Then, Carl Friedrich Gauss developed the theory of Gaussian periods in his Disquisitiones Arithmeticae and formulated a sufficient condition for the constructibility of regular polygons.
The result is known as the Gauss–Wantzel theorem: A positive integer n is of the above form if and only if its totient φ(n) is a power of 2.
Fermat primes are particularly useful in generating pseudo-random sequences of numbers in the range 1, ..., N, where N is a power of 2.
The most common method used is to take any seed value between 1 and P − 1, where P is a Fermat prime.
Very large Fermat primes are of particular interest in data encryption for this reason.
This method produces only pseudorandom values, as after P − 1 repetitions, the sequence repeats.
A poorly chosen multiplier can result in the sequence repeating sooner than P − 1.
An odd prime p is a generalized Fermat number if and only if p is congruent to 1 (mod 4).
If we require n > 0, then Landau's fourth problem asks if there are infinitely many generalized Fermat primes Fn(a).
Because of the ease of proving their primality, generalized Fermat primes have become in recent years a topic for research within the field of number theory.
Besides, we can define "half generalized Fermat numbers" for an odd base, a half generalized Fermat number to base a (for odd a) is
, and it is also to be expected that there will be only finitely many half generalized Fermat primes for each odd base.
If a is a perfect power with an odd exponent (sequence A070265 in the OEIS), then all generalized Fermat number can be algebraic factored, so they cannot be prime.
The number of generalized Fermat primes can be roughly expected to halve as
As in the case where b=1, numbers of this form will always be divisible by 2 if a+b is even, but it is still possible to define generalized half-Fermat primes of this type.
The following is a list of the ten largest known generalized Fermat primes.
