In mathematics, a Fermat number, named after Pierre de Fermat, the first known to have studied them, is a positive integer of the form:${\displaystyle F_{n}=2^{2^{n}}+1,}$ where n is a non-negative integer. The first few Fermat numbers are: 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, ... (sequence A000215 in the OEIS).

If 2^k + 1 is prime and k > 0, then k itself must be a power of 2,^[1] so 2^k + 1 is a Fermat number; such primes are called Fermat primes. As of 2023^[update], the only known Fermat primes are F₀ = 3, F₁ = 5, F₂ = 17, F₃ = 257, and F₄ = 65537 (sequence A019434 in the OEIS).

Basic properties

The Fermat numbers satisfy the following recurrence relations:

${\displaystyle F_{n}=(F_{n-1}-1)^{2}+1}$

${\displaystyle F_{n}=F_{0}\cdots F_{n-1}+2}$

for n ≥ 1,

${\displaystyle F_{n}=F_{n-1}+2^{2^{n-1}}F_{0}\cdots F_{n-2}}$

${\displaystyle F_{n}=F_{n-1}^{2}-2(F_{n-2}-1)^{2}}$

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

${\displaystyle F_{0}\cdots F_{j-1}}$

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 {p_n} is an infinite sequence of distinct primes.

Further properties

• No Fermat prime can be expressed as the difference of two pth powers, where p is an odd prime.
• With the exception of F₀ and F₁, the last digit of a Fermat number is 7.
• The sum of the reciprocals of all the Fermat numbers (sequence A051158 in the OEIS) 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 F₀, ..., F₄ are easily shown to be prime. Fermat's conjecture was refuted by Leonhard Euler in 1732 when he showed that

${\displaystyle F_{5}=2^{2^{5}}+1=2^{32}+1=4294967297=641\times 6700417.}$

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

That 641 is a factor of F₅ can be deduced from the equalities 641 = 2⁷ × 5 + 1 and 641 = 2⁴ + 5⁴. It follows from the first equality that 2⁷ × 5 ≡ −1 (mod 641) and therefore (raising to the fourth power) that 2²⁸ × 5⁴ ≡ 1 (mod 641). On the other hand, the second equality implies that 5⁴ ≡ −2⁴ (mod 641). These congruences imply that 2³² ≡ −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.^[2] 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.^[3] 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^[4])
• Are there infinitely many composite Fermat numbers?
• Does a Fermat number exist that is not square-free?

As of 2024^[update], it is known that F_n is composite for 5 ≤ n ≤ 32, although of these, complete factorizations of F_n are known only for 0 ≤ n ≤ 11, and there are no known prime factors for n = 20 and n = 24.^[5] The largest Fermat number known to be composite is F_18233954, and its prime factor 7 × 2^18233956 + 1 was discovered in October 2020.

Heuristic arguments

Heuristics suggest that F₄ 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 F₅, ..., F₃₂ are composite, then the expected number of Fermat primes beyond F₄ (or equivalently, beyond F₃₂) should be

${\displaystyle \sum _{n\geq 33}{\frac {1}{\ln F_{n}}}<{\frac {1}{\ln 2}}\sum _{n\geq 33}{\frac {1}{\log _{2}(2^{2^{n}})}}={\frac {1}{\ln 2}}2^{-32}<3.36\times 10^{-10}.}$

One may interpret this number as an upper bound for the probability that a Fermat prime beyond F₄ 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.^[6]

Anders Bjorn and Hans Riesel estimated the number of square factors of Fermat numbers from F₅ onward as

${\displaystyle \sum _{n\geq 5}\sum _{k\geq 1}{\frac {1}{k(k2^{n}+1)\ln(k2^{n})}}<{\frac {\pi ^{2}}{6\ln 2}}\sum _{n\geq 5}{\frac {1}{n2^{n}}}\approx 0.02576;}$

in other words, there are unlikely to be any non-squarefree Fermat numbers, and in general square factors of ${\displaystyle a^{2^{n}}+b^{2^{n}}}$ are very rare for large n.^[7]

Equivalent conditions

Let ${\displaystyle F_{n}=2^{2^{n}}+1}$ be the nth Fermat number. Pépin's test states that for n > 0,

${\displaystyle F_{n}}$ is prime if and only if ${\displaystyle 3^{(F_n-<!--\; -\; -->1)/2}\equiv <!--\; \equiv \; -->-<!--\; -\; -->1(\mathrm{mod}{F}_n)}$Zdroj:https://en.wikipedia.org?pojem=Fermat_number
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