Generalized Fermat prime

In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form

:$F_ = 2^ + 1,$

where ''n'' is a non-negative integer. The first few Fermat numbers are:
: 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, ... .

If 2^''k'' + 1 is prime and ''k'' > 0, then ''k'' must be a power of 2, so 2^''k'' + 1 is a Fermat number; such primes are called Fermat primes. , the only known Fermat primes are ''F''₀ = 3, ''F''₁ = 5, ''F''₂ = 17, ''F''₃ = 257, and ''F''₄ = 65537 ; heuristics suggest that there are no more.

Basic properties

The Fermat numbers satisfy the following recurrence relations:

:$F_ = (F_-1)^+1$

:$F_ = F_ \cdots F_ + 2$

for ''n'' ≥ 1,

:$F_ = F_ + 2^F_ \cdots F_$

:$F_ = F_^2 - 2(F_-1)^2$

for ''n'' ≥ 2. Each of these relations can be proved by mathematical induction. From the second equation, we can deduce Goldbach's theorem (named after Christian Goldbach): no two Fermat numbers share a common integer factor greater than 1. To see this, suppose that 0 ≤ ''i'' < ''j'' and ''F''_''i'' and ''F''_''j'' have a common factor ''a'' > 1. Then ''a'' divides both

:$F_ \cdots F_$

and ''F''_''j''; hence ''a'' divides their difference, 2. Since ''a'' > 1, this forces ''a'' = 2. This is a contradiction, because each Fermat number is clearly odd. As a corollary, we obtain another proof of the infinitude of the prime numbers: for each ''F''_''n'', choose a prime factor ''p''_''n''; then the sequence is an infinite sequence of distinct primes.

Further properties

* No Fermat prime can be expressed as the difference of two ''p''th powers, where ''p'' is an odd prime.
* With the exception of ''F''₀ and ''F''₁, the last digit of a Fermat number is 7.
* The sum of the reciprocals of all the Fermat numbers is irrational. (Solomon W. Golomb, 1963)

Primality

Fermat numbers and Fermat primes were first studied by Pierre de Fermat, who conjectured that all Fermat numbers are prime. Indeed, the first five Fermat numbers ''F''₀, ..., ''F''₄ are easily shown to be prime. Fermat's conjecture was refuted by Leonhard Euler in 1732 when he showed that

:$F_ = 2^ + 1 = 2^ + 1 = 4294967297 = 641 \times 6700417.$

Euler proved that every factor of ''F''_''n'' must have the form ''k''2^''n''+1 + 1 (later improved to ''k''2^''n''+2 + 1 by Lucas) for ''n'' ≥ 2.

That 641 is a factor of ''F''₅ can be deduced from the equalities 641 = 2⁷ Ã— 5 + 1 and 641 = 2⁴ + 5⁴. It follows from the first equality that 2⁷ Ã— 5 â‰¡ âˆ’1 (mod 641) and therefore (raising to the fourth power) that 2²⁸ Ã— 5⁴ â‰¡ 1 (mod 641). On the other hand, the second equality implies that 5⁴ â‰¡ âˆ’2⁴ (mod 641). These congruences imply that 2³² â‰¡ âˆ’1 (mod 641).

Fermat was probably aware of the form of the factors later proved by Euler, so it seems curious that he failed to follow through on the straightforward calculation to find the factor. One common explanation is that Fermat made a computational mistake.

There are no other known Fermat primes ''F''_''n'' with ''n'' > 4, but little is known about Fermat numbers for large ''n''. In fact, each of the following is an open problem:
* Is ''F''_''n'' composite for all ''n'' > 4?
* Are there infinitely many Fermat primes? ( Eisenstein 1844)
* Are there infinitely many composite Fermat numbers?
* Does a Fermat number exist that is not square-free?

, it is known that ''F''_''n'' is composite for , although of these, complete factorizations of ''F''_''n'' are known only for , and there are no known prime factors for and . The largest Fermat number known to be composite is ''F''_18233954, and its prime factor was discovered in October 2020.

Heuristic arguments

Heuristics suggest that ''F''₄ is the last Fermat prime.

The prime number theorem implies that a random integer in a suitable interval around ''N'' is prime with probability 1/ln ''N''. If one uses the heuristic that a Fermat number is prime with the same probability as a random integer of its size, and that ''F''₅, ..., ''F''₃₂ are composite, then the expected number of Fermat primes beyond ''F''₄ (or equivalently, beyond ''F''₃₂) should be
:$\sum_ \frac < \frac \sum_ \frac = \frac 2^ < 3.36 \times 10^.$
One may interpret this number as an upper bound for the probability that a Fermat prime beyond ''F''₄ exists.

This argument is not a rigorous proof. For one thing, it assumes that Fermat numbers behave "randomly", but the factors of Fermat numbers have special properties. 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.

Equivalent conditions

Let $F_n=2^+1$ be the ''n''th Fermat number. Pépin's test states that for ''n'' > 0,

:$F_n$ is prime if and only if $3^\equiv-1\pmod.$

The expression $3^$ can be evaluated modulo $F_n$ by repeated squaring. This makes the test a fast polynomial-time algorithm. But Fermat numbers grow so rapidly that only a handful of them can be tested in a reasonable amount of time and space.

There are some tests for numbers of the form ''k''2^''m'' + 1, such as factors of Fermat numbers, for primality.

:Proth's theorem