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The Fermat numbers satisfy the following recurrence relations: = + = + for n ≥ 1, = + = 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.
The quotient is named after Pierre de Fermat. If the base a is coprime to the exponent p then Fermat's little theorem says that q p (a) will be an integer. If the base a is also a generator of the multiplicative group of integers modulo p, then q p (a) will be a cyclic number, and p will be a full reptend prime.
Fermat's Last Theorem (number theory) Fermat's little theorem (number theory) Fermat's theorem on sums of two squares (number theory) Fermat's theorem (stationary points) (real analysis) Fermat polygonal number theorem (number theory) Fernique's theorem (measure theory) Ferrero–Washington theorem (algebraic number theory) Feuerbach's theorem
In number theory, Fermat's Last Theorem (sometimes called Fermats conjecture, especially in older texts) states that no three positive integers a, b, and c satisfy the equation a n + b n = c n for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have infinitely many solutions. [1]
Fermat's factorization method, named after Pierre de Fermat, is based on the representation of an odd integer as the difference of two squares: N = a 2 − b 2 . {\displaystyle N=a^{2}-b^{2}.} That difference is algebraically factorable as ( a + b ) ( a − b ) {\displaystyle (a+b)(a-b)} ; if neither factor equals one, it is a proper ...
In number theory, Fermat's little theorem states that if p is a prime number, then for any integer a, the number a p − a is an integer multiple of p. In the notation of modular arithmetic , this is expressed as a p ≡ a ( mod p ) . {\displaystyle a^{p}\equiv a{\pmod {p}}.}
For integer b > 1, base b may be used if and only if only a finite number of Fermat numbers F n satisfies that () =, where () is the Jacobi symbol. In fact, Pépin's test is the same as the Euler-Jacobi test for Fermat numbers, since the Jacobi symbol ( b F n ) {\displaystyle \left({\frac {b}{F_{n}}}\right)} is −1, i.e. there are no Fermat ...
The works of the 17th-century mathematician Pierre de Fermat engendered many theorems. Fermat's theorem may refer to one of the following theorems: Fermat's Last Theorem, about integer solutions to a n + b n = c n; Fermat's little theorem, a property of prime numbers; Fermat's theorem on sums of two squares, about primes expressible as a sum of ...