The board / number theory / #438
Erdős Problem #438
SOLVEDdatabase updated 2025-08-31
Resolved in some way other than a straight proof or disproof — usual for open-ended questions, or for multi-part problems whose parts went different ways.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
A finite set of natural numbers is square-sum-free if the sum of any two of its elements (possibly equal) is not a square.
The largest size of a square-sum-free subset of `{1, …, N}`.
How large can \(A \subseteq \{1, \ldots, N\}\) be if \(A + A\) contains no square numbers? A problem of Erdős [Er80, Er80c, ErGr80]. Taking all integers \(\equiv 1 \pmod 3\) gives \(|A| \ge N/3\), and Massias observed that all integers \(\equiv 1, 5, 9, 13, 14, 17, 21, 25, 26, 29, 30 \pmod{32}\) give \(|A| \ge \frac{11}{32} N\). Lagarias, Odlyzko and Shearer [LOS83] proved that \(11/32\) is sharp for the modular version of the problem, and Khalfalah, Lodha and Szemerédi [KLS02] proved that it is sharp in general: the maximal such \(A\) satisfies \(|A| \le (\frac{11}{32} + o(1)) N\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved. The authoritative wording is on erdosproblems.com/438.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA363069
Original pageerdosproblems.com/438
Attempts
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