Interactive · The Goldbach comet

The bands you intuited, made visible.

Every even number E is a dot: how far right is E, how high up is its number of Goldbach pairs. Instead of a fuzzy cloud, the dots snap into clean bands — and the color reveals the cause. The more distinct small prime factors E has, the higher its band. The multiples of 3 form the top stripe; the powers of 2 hug the floor. This is the famous Goldbach comet.

Plot even numbers up to

drag max (3000)

color by

up to 1000up to 3000up to 6000up to 10000

0 odd factors (powers of 2) 1 2 3+ (top band)

hover the plot to inspect a number…

Each dot is one even number E: horizontal = E, vertical = how many Goldbach pairs it has. The dots split into bands — and the coloring shows why: numbers with more distinct small prime factors ride higher. The very top band is the multiples of 3 (switch the color mode to see it). Powers of 2 hug the bottom. Y-axis y-value g(E) peaks at 128 in this range.

This is the Hardy–Littlewood prediction drawn in dots: g(E) ≈ 2·C₂ · E/(ln E)² · ∏(q−1)/(q−2) over odd primes q | E. Each small factor multiplies the count — factor 3 nearly doubles it — which is exactly the banding. See first primes for the raw lists behind each dot.