 Manteo2h

2022-06-26

$\frac{4}{3}{x}^{2}+\frac{4}{3}x+1={y}^{2}$
$\frac{8}{3}{x}^{2}+\frac{8}{3}x+1={z}^{2}$ Aaron Everett

Expert

We consider only the general question. It was proved by Matijasevich that there is no algorithm which, on input any Diophantine equation $P\left({x}_{1},{x}_{2},\dots ,{x}_{m}\right)=0$, where $P$ is a polynomial with integer coefficients, will determine whether the equation has an integer solution.
Using a little trick that goes back to Skolem, given any Diophantine equation $P\left({x}_{1},{x}_{2},\dots ,{x}_{m}\right)=0$, we can algorithmically produce a system ${Q}_{i}\left({y}_{1},{y}_{2},\dots ,{y}_{n}\right)=0$, $i=1,\dots ,s$ of quadratic Diophantine equations such that the system has a solution in integers if and only if $P\left({x}_{1},{x}_{2},\dots ,{x}_{m}\right)=0$has a solution in integers.
It follows that there is no algorithm which, given any system of quadratic Diophantine equations, will determine whether the system has a solution in integers.

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