Let $r>0$ be a rational number. There is a criterion for determining whether the equation $x^2+y^2=r$ has a solution in rational numbers, and if we have one solution, then there is a way to generate all solutions. However, how does one find the first solution?
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See also https://math.stackexchange.com/questions/1445058/which-rationals-can-be-written-as-the-sum-of-two-rational-squares – lhf Dec 21 '18 at 11:21
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Thank you, I think this is helpful. – Dec 21 '18 at 11:31
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1@PrakharNagpal This question doesn't ask for the criterion, it asks for how one can find a single solution given that such a solution exists. This is a different question. – Carl Schildkraut Dec 21 '18 at 15:56
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@CarlSchildkraut: You make a valid point, however the current dup target does relate the solution of rational sums of two squares to the integer sum of two squares. For the problem of finding the latter "first solution", see Express integer as sum of two squares. AFAIK one will in general need either a prime factorization or a probabilistic algorithm. If you wanted to elaborate an Answer, it might be best to add it to the integer version of the Question. – hardmath Dec 21 '18 at 22:09
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@CarlSchildkraut I'm sorry, I wasn't paying attention, my bad! – Prakhar Nagpal Dec 22 '18 at 05:15