It is a well-known fact that if an integer is a sum of two rational squares then it is a sum of two integer squares. For example, Cohen vol. 1 page 314 prop. 5.4.9. Cohen gives a short proof that relies on Hasse-Minkowski, but he attributes the theorem (without reference) to Fermat, who didn't have Hasse-Minkowski available. So my question is, how did Fermat prove this theorem? and part 2 of the question is, what is the simplest direct proof? I googled for this result and found a manuscript with a proof that doesn't use Hasse-Minkowski, but it's not very short.
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sums of rational squaresIt is a well-known fact that if an integer is a sum of two rational squares then it is a sum of integer squares. For example, Cohen vol. 1 page 314 prop. 5.4.9. Cohen gives a short proof that relies on Hasse-Minkowski, but he attributes the theorem (without reference) to Fermat, who didn't have Hasse-Minkowski available. So my question is, how did Fermat prove this theorem? and part 2 of the question is, what is the simplest direct proof? I googled for this result and found a manuscript with a proof that doesn't use Hasse-Minkowski, but it's not very short.
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