Erdős Problem #848

Precise statement

Is the maximum size of a set $A\subseteq \{1,\ldots,N\}$ such that $ab+1$ is never squarefree (for all $a,b\in A$) achieved by taking those $n\equiv 7\pmod{25}$? Resolved for all sufficiently large $N$: any near-maximal $A$ is contained in $\{n\equiv 7\pmod{25}\}$ or $\{n\equiv 18\pmod{25}\}$, leaving only a finite check.

The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.

What AI did

GPT-5

Sawhney's note resolving the problem cites a ChatGPT (GPT-5) session in the provenance of the key lemma, and Tao's AI-contributions ledger records the solve as GPT-5 working with Sawhney and Sellke (October-November 2025).

Provider: OpenAI · Prompt public: unknown · Independence: unknown

Verification boundary

unreviewed

erdosproblems.com marks the problem resolved up to a finite check and links Sawhney's note; no formal artifact and no journal review.

Correctness: unknown · statement fidelity: unaudited · peer review: none

Timeline

1. 2025-11-20

   erdosproblems.com/848

   Resolved for all sufficiently large N via a stability theorem; small N remain a finite computation (erdosproblems.com marks the problem DECIDABLE)

Known method families

argument (source-reported)

Source-reported tools: argument.

Independent: unknown · difference confidence: 0

What remains uncertain

VibeMath has not independently audited the mathematical statement, proof, or novelty claim.

• VibeMath has not independently verified the mathematical claim.
• AI-attempt independence and training-data exposure are unknown unless explicitly documented.
• VibeMath has not independently audited the mathematical statement, proof, or novelty claim.