The Oldest Open Problem
In 1742, Christian Goldbach wrote to Leonhard Euler suggesting that every integer greater than 2 could be written as the sum of three primes. Euler reformulated this into the modern 'strong' conjecture: every even number greater than 2 is the sum of two primes. Despite centuries of effort by the greatest minds in mathematics, nobody has proved — or disproved — this deceptively simple statement. This simulation lets you test it yourself for any even number and visualize all prime-pair decompositions.
Counting Prime Pairs
For a given even number 2k, the number of Goldbach representations G(2k) counts distinct pairs (p, q) where p + q = 2k and both are prime. The pair count varies dramatically: 4 has just one pair (2+2), while 100 has six. As numbers grow, G(2k) generally increases, following a trend proportional to 2k/ln²(2k), but with significant fluctuations depending on divisibility by small primes.
The Goldbach Comet
Plotting G(2k) against 2k produces a beautiful scatter pattern called the Goldbach comet. The main body shows the expected growth rate, while distinct 'tails' appear for multiples of 6, 30, and other primorial products — these numbers have systematically more representations because their residues modulo small primes are favorable. The comet structure is a visual fingerprint of the interplay between primes and multiplication.
Computational Frontiers
As of 2014, the conjecture has been verified computationally for all even numbers up to 4 × 10^18 by Oliveira e Silva and collaborators. In 2013, Harald Helfgott proved the weak conjecture (every odd number above 5 is the sum of three primes), a landmark achievement using the Hardy-Littlewood circle method. The strong conjecture, however, may require entirely new mathematical frameworks to settle.