OpenAI publishes GPT-6 Astra result showing infinitely many consecutive primes differ by at most 186
OpenAI publishes GPT-6 Astra result showing infinitely many consecutive primes differ by at most 186
On August 30, OpenAI published a paper on short gaps between primes. The abstract states that GPT-6 Astra produced the proof. The paper was released with a numerical certificate and Lean 4 code that verifies the logical steps in the proof.
The number 186 means that infinitely many pairs of consecutive primes differ by no more than 186. If pₙ is the n-th prime, the theorem states that lim inf (pₙ₊₁ − pₙ) ≤ 186. No matter how far the sequence of primes extends, another such consecutive pair will occur. In 2013, Yitang Zhang established an upper bound of 70 million. The Polymath projects reduced it to 4680, James Maynard reduced it to 600, and the subsequent Polymath project reduced it to 246. The OpenAI paper lowers the bound to 186.
The authors select 40 fixed offsets from a single number n, all between 0 and 186. They prove that, for infinitely many values of n, at least two numbers in the resulting set are prime. Their difference is therefore at most 186.
The proof uses a multidimensional Selberg sieve, a method that assigns numerical weights to such sets and estimates how often they contain primes. The new technique concerns common divisors, which can be factored into three parts of suitable size. This makes it possible to use a broader class of weights in the sieve and obtain the required inequality for the 40 numbers.
The numerical certificate records the finite computation. It uses 77 rational coefficients and interval arithmetic to establish a margin greater than 1/50000. The interval calculations account for rounding errors in a way that preserves the sign of the final inequality.
In August, Axiom Math formalized the previous bound of 246. In that work, Lean verified the chain of reasoning for an already known theorem. In the new repository, the chain begins with three explicitly identified inputs: two analytic estimates and numerical bounds for integrals. Given these inputs, Lean derives the bound of 186. The paper explains the mathematical construction, the certificate verifies its finite computation, and the code allows the derivation from the stated estimates to the theorem to be checked again.