Can MathGPT solve the Twin prime 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

Twin prime conjecture

There are infinitely many primes p with p + 2 also prime.

Exact statement

\#\{p\ \text{prime}: p+2\ \text{prime}\}=\infty,\qquad \text{known: }\liminf_{n\to\infty}(p_{n+1}-p_n)\le 246

There are infinitely many pairs of primes that differ by 2 (such as 3 and 5, 11 and 13, 41 and 43).

Status

Open. Bounded gaps between primes were proved by Yitang Zhang (2013); the best unconditional gap bound is 246.

Posed: Classical; stated in general form by de Polignac (1849).

In plain words

Primes thin out as numbers grow, but pairs like 29, 31 and 1,000,000,007, 1,000,000,009 keep appearing. The conjecture says they never stop. What is proved is weaker: some fixed gap (at most 246) occurs infinitely often.

Key results so far

1919: Brun: the sum of the reciprocals of twin primes converges (Brun's constant ≈ 1.902).
2013: Zhang: infinitely many pairs of primes differ by less than 70,000,000.
2013: Maynard (and independently Tao) give a simpler method, lowering the bound to 600.
2014: The Polymath8b project lowers it to 246 (to 6 assuming a generalized Elliott–Halberstam conjecture).

What our solver can and cannot do

No computer program or AI can solve the Twin prime 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 Twin prime conjecture say?: There are infinitely many pairs of primes that differ by 2 (such as 3 and 5, 11 and 13, 41 and 43).
Has the Twin prime conjecture been solved?: Open. Bounded gaps between primes were proved by Yitang Zhang (2013); the best unconditional gap bound is 246.
Can MathGPT solve the Twin prime 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

Y. Zhang, 'Bounded gaps between primes', Annals of Math. 179 (2014).
J. Maynard, 'Small gaps between primes', arXiv:1311.4600; Annals of Math. 181 (2015).
D.H.J. Polymath, Research in the Mathematical Sciences 1 (2014).

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