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Fundamental Theorem of Arithmetic
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Every integer n > 1 can be written as a product of prime numbers, and this
factorization is unique up to the order of the factors. Equivalently,
there exist primes p1 < p2 < … < pk and positive integers e1, …, ek such that

    n = p1^e1 * p2^e2 * … * pk^ek.

Existence comes from repeatedly dividing n by a prime divisor until only primes
remain; uniqueness follows from Euclid’s lemma (if a prime p divides ab, then
p divides a or b), which forces any two prime factorizations of n to use the
same primes with the same exponents.

For negative integers, include a factor −1; 0 and ±1 are excluded.
