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A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a ...
This theorem is one of the main reasons why 1 is not considered a prime number: if 1 were prime, then factorization into primes would not be unique; for example, = = = … The theorem generalizes to other algebraic structures that are called unique factorization domains and include principal ideal domains , Euclidean domains , and polynomial ...
If it is 1, then n may be prime. If a n −1 (modulo n) is 1 but n is not prime, then n is called a pseudoprime to base a. In practice, if a n −1 (modulo n) is 1, then n is usually prime. But here is a counterexample: if n = 341 and a = 2, then even though 341 = 11·31 is composite.
It will be shown that there exists at least one additional prime number not included in this list. Let P be the product of all the prime numbers in the list: P = p 1 p 2...p n. Let q = P + 1. Then q is either prime or not: If q is prime, then there is at least one more prime that is not in the list, namely, q itself.
Fermat's little theorem states that if p is prime and a is not divisible by p, then a p − 1 ≡ 1 ( mod p ) . {\displaystyle a^{p-1}\equiv 1{\pmod {p}}.} If one wants to test whether p is prime, then we can pick random integers a not divisible by p and see whether the congruence holds.
The similar and still unsolved Legendre's conjecture asks whether for every n ≥ 1, there is a prime p such that n 2 < p < (n + 1) 2. Again we expect that there will be not just one but many primes between n 2 and (n + 1) 2, but in this case the PNT does not help: the number of primes up to x 2 is asymptotic to x 2 /log(x 2) while the number ...
A prime number (or prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. By Euclid's theorem , there are an infinite number of prime numbers. Subsets of the prime numbers may be generated with various formulas for primes .
However, not all numbers of the form with a prime p are prime; for example, 2 11 − 1 = 2047 = 23 × 89 is not a prime number. [ a ] In fact, Mersenne primes are very rare: of the approximately 4 million primes p up to 68,874,199, 2 p − 1 {\displaystyle 2^{p}-1} is prime for only 48 of them.