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where f k is the k-th Fibonacci number. The first condition is the Fermat primality test using base 2. In general, if p ≡ a (mod x 2 +4), where a is a quadratic non-residue (mod x 2 +4) then p should be prime if the following conditions hold: 2 p−1 ≡ 1 (mod p), f(1) p+1 ≡ 0 (mod p), f(x) k is the k-th Fibonacci polynomial at x.
The prime number race generalizes to other moduli and is the subject of much research; ... Note that the first of these obsoletes the ε > 0 condition on the lower bound.
The progressions of numbers that are 0, 3, or 6 mod 9 contain at most one prime number (the number 3); the remaining progressions of numbers that are 2, 4, 5, 7, and 8 mod 9 have infinitely many prime numbers, with similar numbers of primes in each progression.
Any prime number is prime to any number it does not measure. [note 6] Proposition 30 If two numbers, by multiplying one another, make the same number, and any prime number measures the product, it also measures one of the original numbers. [note 7] Proof of 30 If c, a prime number, measure ab, c measures either a or b. Suppose c does not measure a.
Rowland (2008) proved that this sequence contains only ones and prime numbers. However, it does not contain all the prime numbers, since the terms gcd(n + 1, a n) are always odd and so never equal to 2. 587 is the smallest prime (other than 2) not appearing in the first 10,000 outcomes that are different from 1. Nevertheless, in the same paper ...
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}}.}
Cohn's irreducibility criterion is a sufficient condition for a polynomial to be ... If () is a prime number then () is irreducible in []. [1] History and extensions ...
In algebra and number theory, Wilson's theorem states that a natural number n > 1 is a prime number if and only if the product of all the positive integers less than n is one less than a multiple of n.