Erdős Problem #773

What is the size of the largest Sidon subset $A\subseteq\{1,2^2,\ldots,N^2\}$? Is it $N^{1-o(1)}$?

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A question of Alon and Erdős [AlEr85], who proved $\lvert A\rvert \geq N^{2/3-o(1)}$ is possible (via a random subset), and observed that\[\lvert A\rvert \ll \frac{N}{(\log N)^{1/4}},\]since (as shown by Landau) the density of the sums of two squares decays like $(\log N)^{-1/2}$. The lower bound was improved to\[\lvert A\rvert \gg N^{2/3}\]by Lefmann and Thiele [LeTh95].

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Related OEIS sequences: A390813

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Additional thanks to: Akshat Mudgal

When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:

T. F. Bloom, Erdős Problem #773, https://www.erdosproblems.com/773, accessed 2026-02-13