04 — The Prime

2, 3, 5, 7, 11…

divisible only by 1 and itself  —  proven endless around 300 BCE

A prime number can be divided evenly by exactly two things: 1 and itself. Every other whole number breaks apart into primes the way a molecule breaks apart into atoms — primes are the numbers nothing else is built from.

p₁ × p₂ × … × pₙ + 1

Around 300 BCE, the Greek mathematician Euclid proved there is no end to them. Take any finite list of primes, multiply them all together, and add one. The result is never evenly divisible by any prime on that list — so it is either a new prime itself, or it hides one inside it that was never on the list to begin with. No list of primes can ever be complete.

Nature appears to have found the same numbers useful, long before anyone was counting. Two species of cicada spend thirteen or seventeen years underground before emerging all at once, together, and only then. The leading explanation is that a prime-numbered cycle is nearly impossible for a predator or parasite to synchronize with — a twelve-year cycle would collide with anything on a two-, three-, four-, or six-year cycle, but almost nothing shares a cycle of thirteen or seventeen. Other researchers point instead to hybridization between broods, or to ice-age cold slowing the insects' development in the first place. The full explanation is still being worked out.

The same property that makes primes hard to reach also makes them hard to undo. Multiply two enormous primes together and the result is trivial to compute but, so far, extraordinarily hard to factor back apart. That asymmetry is the entire basis of RSA encryption, published in 1978, which still protects a large share of the world's private communication today.

A number that cannot be divided is also very hard to hide inside. Which is exactly where the world keeps its secrets.