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As of January 2016, the largest known prime number has 22,338,618 decimal digits.

There are infinitely many primes, as demonstrated by Euclid around 300 BC.

For example, 5 is prime because 1 and 5 are its only positive integer factors, whereas 6 is composite because it has the divisors 2 and 3 in addition to 1 and 6.

The fundamental theorem of arithmetic establishes the central role of primes in number theory: any integer greater than 1 is either a prime itself or can be expressed as a product of primes that is unique up to ordering. A simple but slow method of verifying the primality of a given number n is known as trial division.

The uniqueness in this theorem requires excluding 1 as a prime because one can include arbitrarily many instances of 1 in any factorization, e.g., 3, 1 · 3, 1 · 1 · 3, etc. It consists of testing whether n is a multiple of any integer between 2 and .

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