# Fundamental theorem o arithmetic

In nummer theory, the fundamental theorem o arithmetic, an aa cried the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater nor 1 either is prime itself or is the product o prime nummers, an that, altho the order o the primes in the seicont case is arbitrary, the primes themselves are nae.[1][2][3] For example,

$1200 = 2^4 \times 3^1 \times 5^2 = 3 \times 2\times 2\times 2\times 2 \times 5 \times 5 = 5 \times 2\times 3\times 2\times 5 \times 2 \times 2 =\cdots\text { etc.} \!$

The theorem is statin twa things: first, that 1200 can be representit as a product o primes, an seicont, na matter hou this is duin, thare will always be fower 2s, ane 3, twa 5s, an na ither primes in the product.

The requirement that the factors be prime is necessary: factorizations containin composite nummers mey nae be unique (e.g. 12 = 2 × 6 = 3 × 4).

## §References

1. Long (1972, p. 44)
2. Pettofrezzo & Byrkit (1970, p. 53)
3. Hardy & Wright, Thm 2