Goldbach’s conjecture is among the oldest and most easily stated unsolved problems in mathematics. It has resisted proof for nearly three centuries.
The statement
Every even integer greater than 2 can be written as the sum of two primes. For example:
The larger the even number, the more representations it tends to have.
History
The conjecture appears in a 1742 letter from Christian Goldbach, a Prussian mathematician, to Leonhard Euler. Goldbach actually wrote a slightly different version; Euler reformulated it into the form we now know and wrote back that he was convinced of its truth but could not prove it.
What is known
- Computationally verified for all even numbers up to .
- Chen’s theorem (1973): every sufficiently large even number is the sum of a prime and a number with at most two prime factors.
- Vinogradov (1937): every sufficiently large odd number is the sum of three primes (the “weak” Goldbach conjecture).
- Helfgott (2013): the weak Goldbach conjecture holds for all odd numbers greater than 5.
The weak conjecture is now a theorem. The strong (original) conjecture remains open.
Why it’s hard
Primes are “hard” because they are defined by what they are not — not divisible by any smaller number except 1. They don’t follow a simple formula. Any approach to Goldbach must understand the additive structure of primes, and this intersection of multiplicative and additive behavior lies at the heart of analytic number theory.
The conjecture has attracted many first-rate mathematicians, but the best tools — the circle method, sieve theory — seem to lack the final kick needed for a proof.
Cultural resonance
Goldbach’s conjecture has appeared in literature and film (notably in the 2007 novel Uncle Petros and Goldbach’s Conjecture). Its simplicity — “every even number is a sum of two primes, right?” — makes it one of the few unsolved problems that can be stated to a child.
Unlike the Millennium Prize problems, Goldbach carries no formal cash reward. Its status is purely mathematical: a long-standing gap in our understanding of the additive structure of the primes.
Frequently asked
How old is this conjecture?
From 1742, in a letter from Christian Goldbach to Leonhard Euler. That makes it one of the oldest unsolved problems in all of mathematics.
How close are we to proving it?
Chen Jingrun proved in 1973 that every sufficiently large even number is the sum of a prime and a number with at most two prime factors. Harald Helfgott proved the weak Goldbach conjecture (every odd number > 5 is a sum of three primes) in 2013. The main conjecture remains open.