Knowra Fermat number Fermat number A Fermat number is an integer of the form 2^(2^n) + 1 for a nonnegative integer n. The first five are prime, while every later known Fermat number is composite.
Exponentiation : Exponentiation is repeated multiplication expressed as a power with a base and exponent. The nested powers in the defining formula make exponentiation the sequence’s basic operation.
Fermat's theorem on sums of two squares : Fermat's theorem on sums of two squares characterizes primes expressible as the sum of two integer squares. Every odd prime divisor of a Fermat number is congruent to 1 modulo a power of two, hence is a sum of two squares.
Fermat prime : A Fermat prime is a prime number of the form 2^(2^n) + 1. The only known examples are the first five Fermat numbers, for n from zero through four.
Constructible polygon : A constructible polygon can be drawn exactly using a straightedge and compass in finitely many steps. Fermat primes appear in the criterion for constructing regular polygons with straightedge and compass.
Prime number : A prime number is an integer greater than one with no positive divisors other than one and itself. Whether a Fermat number is prime is the sequence’s central arithmetic question.
Cyclotomic polynomial : A cyclotomic polynomial is a polynomial whose roots are primitive roots of unity of a specified order. The identity linking consecutive Fermat numbers reflects a factorization of x^(2^m)+1.
Fermat number F5 : Fermat number F5 is 2^32 + 1, equal to 4,294,967,297. Its factorization, 641 × 6,700,417, disproved Fermat’s conjecture that every Fermat number is prime.
Gauss–Wantzel theorem : The Gauss–Wantzel theorem characterizes which regular polygons are constructible with straightedge and compass. Its criterion depends on distinct Fermat primes among the polygon’s prime factors.
Composite number : A composite number is an integer greater than one that has a positive divisor other than one and itself. Most known Fermat numbers are composite, often with factors that reveal their structure.
Fermat factorization : Fermat factorization is the representation of an odd integer as the difference of two squares. It offers a method for finding factors of Fermat numbers when their factors are sufficiently close.
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