Goldbach's Conjecture

Goldbach's Conjecture (1742): Christian Goldbach wrote to Euler: every integer > 2 is the sum of three primes. Euler reformulated: every even integer > 2 is the sum of two primes.

The Goldbach Comet (bottom) shows the number of ways to express each even number as a sum of two primes. It grows roughly as n/(ln n)² — matching the Hardy-Littlewood conjecture. Verified up to 4×10¹⁸ — but no proof exists.