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# Prime Numbers

An integer greater than 1 is prime if its only positive integer divisors are 1 and itself. Otherwise, it is composite. Prime numbers are central elements of number theory, established as such by the fundamental theorem of arithmetic, which recognizes that all integers greater than 1 can be decomposed into unique products of primes. Wolfram|Alpha has many tools for working with primes and related ideas.

Prime Numbers

Check numbers for primality. Generate prime numbers or lists of prime numbers meeting certain conditions.

Determine whether a number is prime:

Specify a prime by its position in the sequence 2, 3, 5, ...:

Generate a list of primes:

Find the nearest prime to a given number:

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Prime Factorization

Decompose numbers into products of primes, which are unique per the fundamental theorem of arithmetic.

Compute a prime factorization:

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Classes of Primes

Work with special subsets of prime numbers. Find numbers within these sets or check for membership.

Find a twin prime pair:

Find a Mersenne prime:

Find one of the few known Fermat primes:

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