MathDB

Erdős Problem 848

Determine whether the following sharp upper-density bound holds: if A⊆NA\subseteq\mathbb{N} satisfies ab+1ab+1 is not squarefree for every a,b∈Aa,b\in A, then d‾(A):=lim sup⁡N→∞∣A∩{1,…,N}∣N≤125\overline{d}(A):=\limsup_{N\to\infty}\frac{|A\cap\{1,\ldots,N\}|}{N}\leq\frac{1}{25}. The bound is sharp, since A={n∈N:n≡7(mod25)}A=\{n\in\mathbb{N}:n\equiv 7\pmod{25}\} satisfies ab+1≡0(mod25)ab+1\equiv 0\pmod{25} for all a,b∈Aa,b\in A.

References

Primary source

GitHub

Additional references

additive-combinatoricsformal-verificationnumber-theory
Posted

Progress summary

Refreshed
Claimed solved

A repository claims a machine-checked proof of the sharp bound, but no independent mathematical audit has confirmed it.

Erdős Problem 848 is an extremal problem whose claimed equality case is the residue class 7(mod25)7 \pmod{25}. The repository presents the result as a formal resolution, while the available reference record treats it provisionally.

Known results

  • Mehtaab Sawhney and Mark Sellke, October--November 2025: the reference wiki records a “Full solution,” associated with GPT-5, but does not determine whether the model proved or assisted with it.

September 2026 formal-proof claim

The repository claims a kernel-checked proof of the sharp extremal bound, with equality at 7(mod25)7 \pmod{25}. No independent audit or peer-reviewed publication was supplied, so the resolution remains unverified.

Current status (as of September 2026): A formal proof and a prior full-solution claim are reported, but Problem 848 is not independently verified as solved.

Sources

Solutions 0

No solutions have been posted yet.