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Math-2790 · mathematics · Number theory

Additive indecomposability of finite modifications of the primes

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For any two subsets A,B of the nonnegative integers, each containing at least two elements, must their sumset differ from the prime set at infinitely many integers?

Short English statement

For any two subsets A,B of the nonnegative integers, each containing at least two elements, must their sumset differ from the prime set at infinitely many integers?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2790-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2790

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 013; 2026-09-24).

Additive indecomposability of finite modifications of the primes

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 013; 2026-09-24), the asserted result is that a set that agrees with the primes apart from finitely many elements cannot be a sumset of two nonnegative-integer sets each having at least two members.

Mathematical statement of the claimed result

Manuscript 1.
Let $\mathcal P$ be the set of primes, and let $A,B\subseteq\mathbb Z_{\ge0}$ with $|A|,|B|\ge2$. For their ordinary sumset $A+B=\{a+b:a\in A,b\in B\}$, the manuscript claims \[ |(A+B)\mathbin{\triangle}\mathcal P|=\infty. \] Consequently even a finite modification of the prime set cannot be expressed as a sum of two nontrivial sets of nonnegative integers; the symmetric difference condition allows finitely many additions or deletions of primes.

Outline of the argument

For manuscript 1 (family 013; 2026-09-24), finite summands are eliminated by shifted-prime sieving. For two infinite summands, complementary residue supports produce character restrictions at separated prime scales; a transfer comparison contracts their possible coverage until it conflicts with coverage of all sufficiently large primes.

Formal verification

For manuscript 1 (family 013; 2026-09-24), OpenAI lists Lean statements for this assertion: a set that agrees with the primes apart from finitely many elements cannot be a sumset of two nonnegative-integer sets each having at least two members.

The central assertion is within the documented formal scope; the additional applications retain their own scope limits.

The linked family Comparator checks are OstmannComplete.lean, OstmannPrimes.lean.

Their named declarations include OAI.Ostmann.inverseGoldbach, OAI.Ostmann.main, OAI.Ostmann.twoInfiniteSummandsImpossible.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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