Goldbach's conjecture

What makes this fascinating

• Every even number, two primes — 4 = 2+2, 100 = 3+97… Is every even number above 2 the sum of two primes?

• Checked into the quintillions — Verified past 4×10¹⁸ with no exceptions, yet proven for none of them.

• Nearly 300 years old — Posed in a 1742 letter to Euler and still wide open today.

Frequently asked questions

What is Goldbach's conjecture?

That every even number greater than 2 can be written as the sum of two prime numbers — for example, 28 = 5 + 23.

Has Goldbach's conjecture been proven?

No. It has been checked by computer past 4 × 10^18 with no exceptions, but a general proof has stayed out of reach for more than 280 years.

Why is it hard?

It connects addition (sums) with the primes, which are defined by multiplication — and bridging those two structures of the integers is notoriously difficult.

More summits in Mathematics

• Riemann Hypothesis

  A 160-year-old pattern in the primes that no one can prove — math's most famous open problem.

• P vs NP

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• Navier–Stokes existence and smoothness

  We use the equations of fluid flow daily — yet can't prove their solutions never blow up.

• Birch and Swinnerton-Dyer conjecture

  A startling claim that a curve's whole-number solutions are encoded in a single function.