PROVED (LEAN)
This has been solved in the affirmative and the proof verified in Lean.
Can every large integer $n$ be written as $n=x^2+y^2-z^2$ with $\max(x^2,y^2,z^2)\leq n$?
The largest integer known which cannot be written this way is $6563$.
[Va99] reports this is 'obvious' if we replace $\leq n$ with $\leq n+2\sqrt{n}$.
This has been resolved in the affirmative by
Chojeckl and GPT-5.4 Pro, who deduce it from a form of Duke's theorem given by Einsiedler, Lindenstrauss, Michel, and Venkatesh
[ELMV12].
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This page was last edited 23 March 2026. View history
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old-bielefelder,
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old-bielefelder
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This problem looks tractable
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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 #1148, https://www.erdosproblems.com/1148, accessed 2026-04-11
