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July 25

Proof of Lagrange's four-square theorem

The ancient Greeks would have had no trouble understanding the statement of this theorem. (Although you would have had to say "four or fewer squares" rather than "four squares", because they did not treat zero as a number.) But, does there exist a proof of this theorem that would have been intelligible to them? Think if you were traveling back in time to Euclid writing his Elements and wanted to give him this theorem so he could put it in with the other theorems about natural numbers. Would it fit? Could it fit? If not, can it be proven that the tools available to him were not adequate to the task (as it has been proven impossible to square the circle) ? 2601:18A:C500:E830:CDC8:78F5:1C09:DC34 (talk) 02:39, 25 July 2023 (UTC)[reply]

It appears to me that the proof using the Hurwitz integers can be rewritten as a proof using only rational numbers by repeatedly unfolding definitions. The result would be a very long and unwieldy proof even using modern algebraic notation. Expressed in the prose form available to Euclid, I think it would not have been possible for even the greatest mathematical genius to wrap their head around it. --Lambiam 07:53, 25 July 2023 (UTC)[reply]

I think they'd appreciate the geometric windmill proof of Fermat's theorem on sums of two squares given in [1] much more. NadVolum (talk) 17:47, 25 July 2023 (UTC)[reply]

In the article it says Diophantus was aware of the theorem, though that's a long way from a proof. Of course Diophantus was half a millennium after Euclid. --RDBury (talk) 20:19, 25 July 2023 (UTC)[reply]