Can MathGPT solve the Goldbach conjecture?

No. No one has a proof, so no solver or AI can give one. MathGPT recognises the problem and shows the exact statement, its status and the established facts, labelled as reference information.

UnsolvedNumber theory

Goldbach conjecture

Every even integer greater than 2 is the sum of two primes.

Exact statement

\forall\, n>2,\ n\ \text{even}:\ \exists\ \text{primes } p,q\ \text{with}\ n=p+q

Every even integer greater than 2 can be written as the sum of two prime numbers.

Status

Open (strong form). Posed in Goldbach's 1742 letter to Euler; verified to 4·10¹⁸. The weak (odd) form was proved by Harald Helfgott in 2013.

Posed: 1742, Christian Goldbach (letter to Euler).

In plain words

4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, 10 = 5 + 5 = 3 + 7, … Large even numbers have many such representations, and the number grows as numbers grow, which is why almost everyone believes it. A proof for all even numbers is still missing.

Key results so far

1937: Vinogradov: every sufficiently large odd number is the sum of three primes.
1973: Chen: every sufficiently large even number is a prime plus a number with at most two prime factors.
2013: Helfgott proves the weak (ternary) Goldbach conjecture: every odd number greater than 5 is the sum of three primes.
2014: Oliveira e Silva, Herzog and Pardi verify the strong conjecture up to 4·10¹⁸.

What our solver can and cannot do

No computer program or AI can solve the Goldbach conjecture by being asked. If you paste it into the MathGPT solver, you get this reference information, labelled as such, never an invented proof. The solver can still compute related special cases and examples. Open the solver.

Frequently asked questions

What does the Goldbach conjecture say?: Every even integer greater than 2 can be written as the sum of two prime numbers.
Has the Goldbach conjecture been solved?: Open (strong form). Posed in Goldbach's 1742 letter to Euler; verified to 4·10¹⁸. The weak (odd) form was proved by Harald Helfgott in 2013.
Can MathGPT solve the Goldbach conjecture?: No. No one has a proof, so no solver or AI can give one. MathGPT recognises the problem and shows the exact statement, its status and the established facts, labelled as reference information.

References

H. Helfgott, 'The ternary Goldbach problem', arXiv:1312.7748.
T. Oliveira e Silva, S. Herzog, S. Pardi, Math. Comp. 83 (2014).

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