GPT 5.6 manuscript proposes a new method for constructing long gaps between primes

GPT 5.6 manuscript proposes a new method for constructing long gaps between primes

A 48-page manuscript dated 25 August is attributed to GPT 5.6 Sol. On 26 August, the user DottedCalculator submitted a related partial proof claim that identifies the model as GPT 5.6 Pro. Both sources describe a new approach to the classical problem of large gaps between consecutive primes.

Long gaps between primes can be constructed using divisibility. For each small prime, one selects a residue class. Every number in that class is divisible by the corresponding prime. The Chinese remainder theorem combines all these conditions into a single shift, producing an interval in which every number is composite. Covering more numbers in this way guarantees a longer gap between consecutive primes.

The manuscript changes the first stage of this construction. For some primes, it selects the zero residue more often, which makes each number in the chosen class divisible by that prime. As a result, composite numbers containing these prime factors are more likely to be covered at the first stage. For larger primes, the method selects residue classes containing many composite numbers that remain uncovered after this stage, so one residue class covers an entire group.

The remaining primes are handled by another part of the framework from the 2018 paper by Ford, Green, Konyagin, Maynard, and Tao. A Maynard weight identifies residue classes containing several primes, and a covering theorem selects a collection with limited overlap. The Chinese remainder theorem then converts the complete covering into an interval of composite numbers.

The manuscript claims the bound G(T) ≥ c · log T · log₂T / log₄T for sufficiently large T. Here, G(T) is the largest gap between consecutive primes whose right endpoint is at most T, and the subscripts denote iterated logarithms. The problem page compares this formula with the 2018 bound. The claimed improvement grows with T, although only through iterated logarithms.

For a different problem concerning gaps between primes, Lean has already verified every formalized step. The file for the present problem contains the theorem statement, but the proof body is marked sorry.

When GPT-5.6 proposed an approach to the Croisot conjecture, the conjecture's author and an independent group subsequently checked the proof in detail. The forum user bengreen wrote that a group is rewriting the present argument, including the shifted preliminary sieve, separate coverings of composite numbers and primes, and the conversion of the covering into a prime gap.