The primes are the integers that can't be divided — 2, 3, 5, 7, 11, 13, and onwards. That simple constraint, applied consistently to every number, produces a distribution that has resisted complete understanding for two thousand years.
We know they thin out. The prime number theorem says that near N, primes appear roughly once every ln(N) integers. By the time you reach ten million, you'd expect a prime about every sixteen numbers. But the gaps are irregular — sometimes two primes appear next to each other (twin primes), sometimes there's a long desert. Whether there are infinitely many twin primes is still unknown.
We know the Riemann hypothesis says something deep about how evenly they're distributed — if it's true, the primes deviate from their average density in a tightly bounded way. It's been unproven since 1859 and is probably the most important unsolved problem in mathematics. All the prizes, all the effort.
What strikes me is how useful they are despite this incompleteness. RSA encryption works because multiplying two large primes is easy, but factoring the product is hard — and no one can prove why, exactly. We build the security of the internet on a fact we understand empirically but not theoretically. The gap between "it works" and "we know why it works" can be enormous, and we build on it anyway.
The primes also appear unexpectedly in other domains. The cicadas that emerge every 13 or 17 years (both prime) are hypothesized to do this to avoid coinciding with the cycles of predators with shorter periods. The mathematics of avoidance. Evolution found the Sieve of Eratosthenes.
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