what you're seeing
Every integer 1, 2, 3, … laid out in a square spiral; only the primes are lit. The dots
fall onto diagonals all by themselves — and each diagonal is exactly a quadratic
4t² + bt + c. Two independent oracles — a Sieve of Eratosthenes and a
from-scratch trial-division check — must agree on every number up to N, so no lit dot can lie.
the punchlines
The brightest diagonal is Euler's n²−n+41: it lands on a prime for n = 0, 1, 2, … all the
way to 40 — forty-one primes in a row, an unbroken gold streak. Then at
n = 41 it gives 41² = 1681, its first composite. Pick it and watch the streak run, and break,
exactly there.
The running count of lit cells is π(N) — exact, sieve-counted. As N grows it tracks
N / ln N, the Prime Number Theorem's estimate; the ratio tightens toward 1. The primes
thin out, but never quite stop.