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According to the Wolfram website, Mathematica can solve arbitrary bivariate quadratic equations in the integers.

However, when I run

Reduce[-7102 + 1525 x + 1525 x^2 - 25620 y + 1525 x y - 22875 y^2 == 0, {x, y}, Integers]

it does not output an answer after a significant time waiting. (I'm currently using Mathematica version 13.0.) On the other hand, if I use this website, it tells me immediately that there are no integer solutions.

Is there a working implementation of an algorithm, in Mathematica, to solve arbitrary bivariate quadratic integer equations over the integers? (I'd even be satisfied with an algorithm telling me whether or not there are any integer solutions.)

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  • $\begingroup$ Do you have / know of an algorithm to achieve what you want? Perhaps that could be easily implemented in Mathematica. $\endgroup$
    – MarcoB
    Apr 14 at 22:34

1 Answer 1

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In this case, at least, Mathematica can show that there are no solutions using modular arithmetic e.g.

Reduce[-7102 + 1525 x + 1525 x^2 - 25620 y + 1525 x y - 
   22875 y^2 == 0, {x, y}, Modulus -> 5]
(* False *)

By inspection, it is clear that the LHS reduces to 3 mod 5.

I don't know whether this approach can be generalised sufficiently to help you.

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