Goldbach conjectured (1742) that every even integer ≥ 4 is the sum of two primes. For each even n, plot the number of ways: g(n). The cloud of points forms a comet whose internal bands betray the deep arithmetic of small primes.
g(n) isn't smooth in n — it's strongly influenced by n's prime factorization. Even numbers whose factorization includes many small primes have more Goldbach representations: 30 = 2·3·5 has more pairs than 28 = 4·7. So the comet stratifies. The lowest band corresponds to even n with smallest prime factor 2 only (n ≡ 2 mod 4 with no other small factors); higher bands correspond to numbers with additional small prime divisors. Color by smallest prime factor of n/2 to see the strata segregate by color.
n / ln²(n) by the Hardy-Littlewood conjecture — meaning even at very large n, there are more representations, not fewer. The Goldbach conjecture is unproven but verified up to ~4·10¹⁸.