Prime-divisor counts among pairwise sums

Let f(n) be the least number of distinct primes dividing the pairwise sums of an n-element set of positive integers. Must f(n)/log n tend to infinity?

Why it matters and what counts as progress

Why it matters: The question asks how many different primes must be revealed by addition inside a finite set of integers.

What counts as progress: Prove a superlogarithmic lower bound, strengthen it to a polynomial bound or find constructions limiting the possible exponent.

How the problem developed

1934 · Initial bound [1][2]
Paul Erdős and Pál Turán Introduced the pairwise-sum prime-divisor extremal problem and proved logarithmic lower and n/log n upper bounds.
6 Sept 2026 · Lean-checked resolution [2][3]
GPT-6 Astra (FrontierMath Erdős benchmark); Tom Adamczewski maintains the proof repository A Lean-checked proof establishes f(n)/log n→∞, resolving Erdős's qualitative conjecture. The repository includes separate polynomial lower-bound arguments, including a square-root bound.

AI contribution

GPT-6 Astra proves a superlogarithmic bound for Erdős Problem 126

Let f(n) be the minimum number of distinct prime divisors among pairwise sums of any n-element subset of positive integers. The compared theorem proves f(n)/log n→∞. Separate Lean resolutions give stronger polynomial lower bounds, including a square-root bound.

AI involvementAI-autonomous

ImportanceNotable

Technical verificationFormal (Lean 4 with Mathlib and Comparator)

Human understandingPartial