OpenAI research collection · 1 manuscript · Source checked 2026-10-19
The idea in plain language
If the sum of the reciprocals of a set of positive integers keeps growing without bound, this paper claims the set must contain evenly spaced sequences of every finite length. It also gives quantitative bounds for sets that avoid such patterns.
Why it matters
This links a simple measure of abundance to unavoidable structure, while the quantitative theorem supplies more detail than the headline alone.
Read the technical source claim
Proves Erdős's conjecture that every set of positive integers with divergent reciprocal sum contains arithmetic progressions of every finite length. Quantitatively, for each fixed k ≥ 3, every subset of \{1,\ldots,N\} with no nonconstant k-term progression has size at most C_kNexp[-c_k(log N)^{\varepsilon_k}], with positive constants depending only on k.
Read the manuscripts
Short excerpts below come from OpenAI's abstracts. The full abstract and original PDF preserve the paper's precise scope.
MANUSCRIPT 01
Quasipolynomial Bounds for Arithmetic Progressions
We prove Erdős's conjecture that every set of positive integers with divergent reciprocal sum contains arithmetic progressions of every finite length. More quantitatively, for every fixed k ≥ 3, we show \displaystyle r_k(N)≤ C_kNexp\bigl(-c_k(log N)^{\varepsilon_k}\bigr) with C_k,c_k,\varepsilon_k> 0, where r_k(N) is the…
Full source abstract
We prove Erdős's conjecture that every set of positive integers with divergent reciprocal sum contains arithmetic progressions of every finite length. More quantitatively, for every fixed k ≥ 3, we show \displaystyle r_k(N)≤ C_kNexp\bigl(-c_k(log N)^{\varepsilon_k}\bigr) with C_k,c_k,\varepsilon_k> 0, where r_k(N) is the largest size of a subset of \{1,\ldots,N\} with no nonconstant k-term arithmetic progression.
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New mathematical reasoning could help AI tackle harder scientific and computational problems. How reliably these skills transfer beyond mathematics will determine how much they contribute to general intelligence.