Extremal sets with every product-plus-one nonsquarefree
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How large can A⊂{1,…,N} be if ab+1 is nonsquarefree for every a,b∈A? The conjectured asymptotic extremal examples are particular residue classes modulo 25.
Why it matters and what counts as progress
Why it matters: The problem connects a squarefree obstruction for every pair to the size and modular structure of an extremal integer set.
What counts as progress: Prove a matching size upper bound, classify all extremal residue structures or obtain an effective threshold in N.
How the problem developed
- 1992 · Problem [1]
Paul Erdős and András Sárközy Asked for the maximum size of A⊆[N] such that ab+1 is nonsquarefree for every a,b∈A, and whether the residue class 7 mod 25 is extremal.
- 2025 · Asymptotic result [2]
Mehtaab Sawhney, with GPT-5 Pro assistance and prior ideas from Wouter van Doorn, Desmond Weisenberg and Zachary Cambie Proved that for all sufficiently large N the 7 mod 25 class has maximum size, with a near-extremal structural statement involving the 7 and 18 mod 25 classes.
AI contribution
On A ⊆ [N] Such That ab + 1 Is Never Squarefree for a, b ∈ A
For sufficiently large N, Sawhney bounds the size of A⊂[N] with ab+1 nonsquarefree for every a,b∈A by the size of the residue class 7 mod 25, and identifies the extremal 7 and possibly 18 mod 25 classes. The model-origin part is a squarefree-density sieve with prime exclusions depending on b and the progression difference.
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