Concept
Every even integer > 2 is the sum of two primes. Conjectured 1742; verified to 4×10¹⁸ but unproven for 280 years.
Understand it in one breath
The simplest-looking conjectures stay open the longest. 4=2+2, 6=3+3, 100=3+97=11+89=…; computers have verified up to 4×10¹⁸ with no counterexample. Yet no one has proved it for all even integers. The weak version (every odd ≥ 7 is a sum of three primes) was finally settled in 2013 by Helfgott — 76 years after Vinogradov's 1937 result for sufficiently large odd numbers.
At a glance
Even integer n | Number of representations as a sum of primes | Example |
|---|---|---|
4 | 1 | 2 + 2 |
10 | 2 | 3 + 7, 5 + 5 |
100 | 6 | 3 + 97, 11 + 89, 17 + 83, 29 + 71, 41 + 59, 47 + 53 |
1,000 | 28 | 3 + 997, 11 + 989, … |
10,000 | 127 | The number of representations grows explosively |
4 × 10¹⁸ | Largest verified bound | Zero counterexamples — every case checked by computer |
The number of representations grows so large that reaching zero seems impossible. Yet there is no proof for every n — the conjecture has remained unsolved for 280 years.
Key formula
Modern applications
A proving ground for research on the distribution of primes and for analytic number theory.
Beyond MathVoyage
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