Fermat number

Definition and basic properties

The nth Fermat number is given by the closed form F_n = 2^{2^n} + 1. The sequence begins with small values that grow extremely rapidly. Two fundamental properties are often highlighted:

• Pairwise coprimality: any two distinct Fermat numbers are relatively prime. Equivalently, for every n ≥ 1 one has F_0 F_1 · … · F_{n-1} = F_n - 2, so each new Fermat number exceeds the product of all earlier ones by exactly 2.
• Fermat primes: if F_n itself is prime it is called a Fermat prime. A related fact is that a prime divisor p of a number of the form 2^m + 1 must satisfy certain congruence conditions that force m to be a power of two when the resulting number is prime.

Examples and known factorizations

The first few Fermat numbers are explicitly:

1. F_0 = 2^{1} + 1 = 3
2. F_1 = 2^{2} + 1 = 5
3. F_2 = 2^{4} + 1 = 17
4. F_3 = 2^{8} + 1 = 257
5. F_4 = 2^{16} + 1 = 65537

These first five are prime. Later Fermat numbers become composite; for example

• F_5 = 2^{32} + 1 = 4294967297 = 641 × 6700417
• F_6 = 2^{64} + 1 = 18446744073709551617 = 274177 × 67280421310721
• F_7 and F_8 are also composite and admit known factorizations with large prime factors.

Many researchers maintain updated factor lists; complete factorizations of several Fermat numbers have been published and collected in factor tables and online resources such as the database at Prime Factors of Fermat Numbers.

History and significance

Pierre de Fermat observed the first few terms and conjectured that all numbers of the form 2^{2^n}+1 were prime. This belief held until Euler showed that F_5 is composite by providing the factor 641, disproving Fermat's conjecture. The study of Fermat numbers has influenced development in number theory, computational factoring, and the search for large primes.

Applications and related results

Fermat primes are tied to classical geometry: a regular polygon with n sides is constructible with straightedge and compass precisely when n is a power of two times a product of distinct Fermat primes. This criterion explains why the regular 17-gon (since 17 = F_2) is constructible and why constructibility questions connect to the existence of Fermat primes.

Open questions and distinctions

Only a handful of Fermat numbers are prime (F_0 through F_4). No other Fermat primes are currently known, and it is unknown whether infinitely many Fermat primes exist. The rapid growth of F_n makes testing and factoring large indices computationally demanding. Researchers continue to seek factors, test primality, and explore the arithmetic properties of these numbers; see introductory and technical references for further reading on primality conditions and congruences for divisors of numbers of the form 2^m+1 (prime-related results) and general sequence entries (sequence notes).