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One of Euler's discoveries was if an integer $n$ can be represented as a sum of two squares in two distinct ways, then one can factor $n$ explicitly.

Of course, the method was ineffective as an algorithm, but I was wondering if anyone knew of a similar method for factoring integers using a sum of four squares representation.

JasonM
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    Don't see any relevance of four squares. Meanwhile, Euler's method generalizes to $a x^2 + b y^2,$ and was used for factoring by the Lehmers, father and son, in the 1930's – Will Jagy Jun 01 '16 at 04:24
  • I ask because Euler's method was basically a reversal of the Brahmagupta-Fermat identity, and perhaps one could reverse Euler's four square identity instead. – JasonM Jun 01 '16 at 05:49
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    You can factor integers with the sum of 4 squares but it does not involve a nice and simple formula like in the case of Euler's method. You first express $N=pq=a^2+b^2+c^2+d^2$ as a sum of 4 squares then you combine the squares $c=(a^2+b^2, a^2+b^2+c^2...)$ or $c=(a+b, a+b+c...)$ and you calculate the $gdc (N,c)$. The details can be found here: https://math.stackexchange.com/questions/3088664/can-factoring-with-the-sum-of-4-squares-be-made-more-efficient – user25406 Jan 31 '19 at 19:40

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