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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.
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 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.
Cunningham numbers of the form b n − 1 can only be prime if b = 2 and n is prime, assuming that n ≥ 2; these are the Mersenne numbers. Numbers of the form b n + 1 can only be prime if b is even and n is a power of 2, again assuming n ≥ 2; these are the generalized Fermat numbers, which are Fermat numbers when b = 2.
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]
In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction [1] used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for a number, then the same would be true for a smaller number, leading to an infinite descent and ultimately a contradiction. [2]
For these numbers, repeated application of the Fermat primality test performs the same as a simple random search for factors. While Carmichael numbers are substantially rarer than prime numbers (Erdös' upper bound for the number of Carmichael numbers [ 3 ] is lower than the prime number function n/log(n) ) there are enough of them that Fermat ...
Fermat's method works best when there is a factor near the square-root of N. If the approximate ratio of two factors ( d / c {\displaystyle d/c} ) is known, then a rational number v / u {\displaystyle v/u} can be picked near that value.