Frontier Science › Mathematics · Feature · Practical STEM, Part 3
Goldbach's guess: why one unsolved problem from Königsberg still matters
In 1742, Königsberg-born Christian Goldbach wrote to Euler with a hunch about prime numbers. Nearly three centuries later, nobody has proved it — and the hunt has sharpened the mathematics that keeps our data safe.
Pick any even number bigger than 2. Now write it as the sum of two primes — numbers divisible only by 1 and themselves. 4 is 2 + 2. 10 is 3 + 7. 100 is 47 + 53. It always seems to work.
A letter between friends
Christian Goldbach was born in Königsberg in 1690. He travelled widely, worked in St Petersburg and corresponded with the great mathematicians of his day — above all, Leonhard Euler. In a letter dated 7 June 1742, he suggested that whole numbers could always be written as sums of primes in a certain way. Euler replied with a cleaner version, the one we know today: every even number greater than 2 is the sum of two primes. Euler was confident it was true. He could not prove it. Nor has anyone since.
Checked to the stars, proved to nobody
Computers have tested the conjecture for every even number up to 4 × 10¹⁸ — four billion billion — without a single failure. Yet in mathematics, evidence is not proof. A pattern can hold for a trillion cases and break on the next one.
There has been real progress. In 2013, the mathematician Harald Helfgott proved the “weak” Goldbach conjecture: every odd number greater than 5 is the sum of three primes. The original, “strong” version remains open.
Why an unsolved problem is useful
Hard problems are training grounds. The search for a proof has pushed mathematicians to invent new ways of counting and estimating primes. Facts about primes also protect us every day. When you pay online or send a private message, your security depends on them — for example, that multiplying two large primes is easy, but splitting the result back into its primes is extremely hard.
The practical lesson
Goldbach’s conjecture shows the value of a good question. It is simple enough for a child to understand and deep enough to defeat experts for nearly 300 years. The best problems in science, engineering and business are like that: easy to state, hard to solve, and rewarding to chase — because of everything you learn along the way.
Try it yourself
Write the even numbers from 4 to 50 and find two primes that add up to each one. Which numbers have the most ways? You are repeating, by hand, the first steps of a hunt that has lasted since 1742.
Sources: Goldbach’s letter to Euler, 7 June 1742 · Oliveira e Silva, Herzog and Pardi, “Empirical verification of the even Goldbach conjecture,” Mathematics of Computation (2014) · Helfgott, “The ternary Goldbach conjecture is true,” arXiv (2013) · Rivest, Shamir and Adleman, Communications of the ACM (1978).
How this story was made: drafted by Luise Hönig (Virtual Contributor), completed by Gustav Hohenstein, approved by Yoshihiro Horiuchi, Editor-in-Chief.
How this story was made: drafted by Luise Hönig (Virtual Contributor), completed by Gustav Hohenstein, approved by Yoshihiro Horiuchi, Editor-in-Chief.