Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 431

claims/: The 1 claim page of Problem 431, one per claimant's result; the problem's standing derives from them.

Statement. Are there two infinite sets $A$ and $B$ such that $A+B$ agrees with the set of prime numbers up to finitely many exceptions?

Status. Disproved here; the site's label is OPEN (page last edited 8 April 2026, no proof claim recorded there as of 2026-10-06); its commentary expects the answer no and records the Elsholtz--Harper square-root bounds on a hypothetical decomposition as the best result in that direction. One accepted full claim is recorded on the OpenAI release's claim page: a manuscript of 24 September 2026 proves Ostmann's inverse Goldbach conjecture, that no sumset $A+B$ with $|A|,|B|\ge2$ differs from the primes in finitely many elements, which answers the question in the negative. Its Lean proofs, of that conjecture and of the case of two infinite summands that the question asks about, were built by this corpus with only the three standard axioms, their fingerprints identical to the release's comparator challenges, and the statement audit found the latter exactly the question with the answer no over the nonnegative integers. The claim has no outside review or refereed publication; the frontmatter standing is the accepted claim's.

Source. erdosproblems.com/431, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #431, https://www.erdosproblems.com/431.

References.

[El01] Elsholtz, Christian, The inverse Goldbach problem. Mathematika (2001), 151-158.
[ElHa15] Elsholtz, Christian and Harper, Adam J., Additive decompositions of sets with restricted prime factors. Trans. Amer. Math. Soc. (2015), 7403-7427.
[Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.
[Gr90] Granville, Andrew, A note on sums of primes. Canad. Math. Bull. (1990), 452-454.
[TaZi23] Tao, Terence and Ziegler, Tamar, Infinite partial sumsets in the primes. J. Anal. Math. (2023), 375-389.

Formalization. Statement in formal-conjectures. The release's declaration OAI.Ostmann.twoInfiniteSummandsImpossible proves the Statement with the answer no for sets of natural numbers, and OAI.Ostmann.inverseGoldbach and OAI.Ostmann.main prove the stronger inverse Goldbach form; this corpus built all three from the pinned revision with the axioms propext, Classical.choice and Quot.sound only, as the claim page records.

Progress

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Linked library material

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Linked from (17)

OpenAI: the primes are additively indecomposable up to finite changes problems/primes/E0431/claims Primes primes/elsholtz_2001_inverse_goldbach_problem Corollary (p. 2): no A+B+C with |A|,|B|,|C| >= 2 agrees with the primes up to finitely many elements Theorem (pp. 1-2): x^{1/2}/(log x)^5 << A(x) << x^{1/2}(log x)^4 when A+B agrees with the primes primes/elsholtz_2015_additive_decompositions_sets_restricted_prime_factors Theorem 2.6 (p. 6): summands of a hypothetical asymptotic decomposition of the primes have size about x^(1/2)primes/granville_1990_note_sums_primes Theorem (p. 452): conditional infinite sets of primes with prime weighted sums primes/openai_2026_additive_indecomposability_primes Theorem 1.1: (A+B) △ P is infinite whenever |A|, |B| ≥ 2 Theorem 2.3: no two infinite sets sum to the primes up to a finite set primes/tao_2023_infinite_partial_sumsets_primes Corollary 1.6: increasing sequences (a_i), (b_j) with a_i + b_j prime whenever 1 <= i < j Theorem 1.3: under Dickson-Hardy-Littlewood, an infinite set B of primes with b + b' + 1 prime for distinct b, b'Theorem 1.5: an infinite sequence h_1 < h_2 < ... all of whose initial k-tuples are prime-producing

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