The Atoms of Arithmetic

This article records tradition as it has been passed down and reported. Its sources are not yet part of the atlas's verified catalogue.

Chemistry has its periodic table because every substance breaks down into a fixed, finite list of elements combined in different ratios. Arithmetic has an equivalent, and it is older by more than two thousand years: every whole number greater than one is built from primes, multiplied together, in exactly one way. Twelve is two times two times three and nothing else; there is no second recipe. Euclid proved, in Book IX of the Elements, that the supply of these atoms never runs out, with an argument that remains one of the cleanest in all of mathematics: assume there is a longest possible list of primes, multiply them all together and add one, and the new number must either be prime itself or divisible by a prime not on the supposed complete list, either way contradicting the assumption. What Euclid could not have anticipated is how strange the primes would turn out to be at larger scales. No formula predicts where the next one falls; twin primes, pairs two apart like 11 and 13, keep appearing at every scale mathematicians have checked, and nobody has proved they do not eventually stop (the atlas's own Twin Prime Conjecture is exactly that unresolved question). And in the twentieth century, the very unpredictability that made primes a matter of pure curiosity for two millennia became the foundation of a very practical problem: modern encryption relies on the fact that multiplying two large primes together is easy, and working backward from the product to find the original primes is, for now, hard enough to keep secrets safe.