The reason 1 is not a prime number is a theorem about uniqueness
Every whole number above 1 breaks into primes in exactly one way, apart from the order. 1200 always contains four 2s, one 3 and two 5s. If 1 counted as prime, you could tack on as many 1s as you liked and uniqueness would collapse, which is a key reason mathematicians exclude it.
The fundamental theorem of arithmetic, also called the unique factorisation theorem, says that past 1, every whole number is a prime or splits into primes in only one way, ignoring order. With the convention that an empty product equals 1, it can be stated for every positive integer. The primes matter: factorisations allowed to include composite numbers need not be unique. Writing a number as a product of increasing prime powers gives its canonical or standard form.
The ideas go back to Euclid's Elements. Book VII, Proposition 30, now called Euclid's lemma, says that if a prime divides a product of two numbers it must divide at least one of them, and it is the key step in proving uniqueness. Propositions 31 and 32 show every number has a prime factor and can therefore be broken into primes. Book IX, Proposition 14 proves uniqueness only when no prime repeats, a limitation André Weil pointed out.
Credit for the full statement goes further east and later. Kamāl al-Dīn al-Fārisī is described as the first to state the theorem, finishing the path Euclid began, and Article 16 of Carl Friedrich Gauss's Disquisitiones Arithmeticae appears to contain the first proof that the factorisation is unique.
The theorem extends to structures called unique factorisation domains, which include polynomial rings over a field and Euclidean domains. It fails, however, for many rings of algebraic integers, and that failure helps explain why Fermat's Last Theorem was so hard. Many of the false proofs offered during the 358 years between Fermat's claim and Andrew Wiles's proof quietly assumed unique factorisation where it does not hold.
Source: Fundamental theorem of arithmetic
