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According to the [[fundamental theorem of arithmetic]], proven by [[Carl Friedrich Gauss]], every positive integer has a unique factorization into prime numbers.

According to the [[fundamental theorem of arithmetic]], proven by [[Carl Friedrich Gauss]], every positive integer has a unique factorization into prime numbers.

This means that every integer larger than 1, can be expressed as a product of one or more [[prime|primes]] in one unique way. For example, 132 = 2 * 2 * 3 * 11. There is no other product of primes that equals 132.

This means that every integer larger than 1, can be expressed as a product of one or more [[prime|primes]] in only one way. For example, 132 = 2 * 2 * 3 * 11. There is no other product of primes that equals 132.

There is no effective algorithm for finding a certain integer's unique factorization, and finding the prime factors for large numbers can take considerable time (millions of years) even with the most advanced computers. This is referred to as the prime factorization problem, and it is believed to be NP-complete.

There is no effective algorithm for finding a certain integer's unique factorization, and finding the prime factors for large numbers can take considerable time (millions of years) even with the most advanced computers. This is referred to as the prime factorization problem, and it is believed to be NP-complete.