An elementary quantitative proof of Erdős Problem 126
Abstract
Let 𝐴 be a finite set of distinct positive integers, and let 𝑆(𝐴) be the set of primes dividing a sum of two distinct elements of 𝐴. We give an elementary quantitative proof of |𝐴| ≤ 3|𝑆(𝐴)|2 for |𝐴| ≥ 2, which implies the superlogarithmic growth asked for in Erdős Problem 126. The construction originates in an AI-generated formal proof. Prime colours orient negation orbits modulo prime powers; their supports form laminar families, and logarithmic weights relate sums to differences. A positive semidefinite correction accounts for self-opposite residues and transfers conditional negative type from the kernel log(𝑥 + 𝑦) to an arithmetic kernel. Applying the signed laminar-kernel theorem from a companion paper then gives the cardinality bound. We provide the arithmetic and analytic details, explain how they fit together, and record the provenance of the formal construction in a short appendix.
Keywords:
- Keyword: Pairwise sums; prime support
- Keyword: Erdős Problem 126
- Keyword: prime-power congruences
- Keyword: laminar families
- Keyword: conditionally negative kernels
How to Cite:
Bao, S., (2026) “An elementary quantitative proof of Erdős Problem 126”, Intelligence, Mathematics and Society 1(1), 1–16.
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