The board / number theory / #431
Erdős Problem #431
inverse Goldbach problem
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
Are there two infinite sets \(A\) and \(B\) such that \(A+B\) agrees with the primes up to finitely many exceptions? This was disproved by an internal OpenAI model in September 2026 [OAI26a]; see the linked Lean proof.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/431.Newer public claim (6 Oct 2026): OpenAI's internal-model release claims, for this problem, Ostmann: no finite modification of the primes is a sumset A+B — family 13 at github.com/openai/math, Lean-formalized. Bloom calls these claims, not endorsements; anything on this page that says the question is open was written before that release and has not re-checked it.
Prizenone
Topicsnumber theory, primes
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/431
Attempts
submit one →Read our working report →
The problem remains open; verified current progress is Croot--Yip's July-2026 quadratic-intersection theorem, and this report adds an exhaustive sharp computation of the balanced all-prime-sum paramet
PARTIAL