Fermat prime
Definition
A Fermat prime, named after the French mathematician Pierre de Fermat, is a positive integer of the form F_n = 2^(2^n) + 1, where n is a nonnegative integer. The concept of Fermat primes is a significant topic in the field of number theory, a branch of pure mathematics devoted to the study of the integers.
Properties
Fermat primes have several unique properties that distinguish them from other types of primes.
Primality
The most fundamental property of Fermat primes is their primality. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. However, not all numbers of the form F_n = 2^(2^n) + 1 are prime. In fact, Fermat himself erroneously believed that all numbers of this form were prime, which is not the case. The only known Fermat primes are F_0 = 3, F_1 = 5, F_2 = 17, F_3 = 257, and F_4 = 65537.
Regularity
Fermat primes are also regular primes, a property that has implications in the field of algebraic number theory. A regular prime is a prime number p such that the p-th Bernoulli number B_p is not divisible by p. The concept of regular primes is central to the theory of cyclotomic fields and the Kummer's theorem.
Constructibility
Another important property of Fermat primes is their connection to constructible polygons. A regular polygon with n sides is constructible using compass and straightedge alone if and only if n is the product of a power of 2 and any number of distinct Fermat primes. This property is a consequence of the Gauss-Wantzel theorem, which provides a complete characterization of constructible polygons.
History
The study of Fermat primes has a rich history, dating back to the 17th century. Pierre de Fermat first proposed the conjecture that all numbers of the form F_n = 2^(2^n) + 1 are prime in a letter to his friend and fellow mathematician Marin Mersenne in 1640. However, it was not until 1732 that the Swiss mathematician Leonhard Euler disproved Fermat's conjecture by showing that F_5 = 2^(2^5) + 1 is not a prime number.
Open Problems
Despite centuries of research, many questions about Fermat primes remain unanswered. The most famous of these is the Fermat prime conjecture, which asks whether there are infinitely many Fermat primes. This conjecture, which remains one of the oldest unsolved problems in number theory, has motivated much of the research in the field.