<s>[INST] Show that no non-zero integers satisfy this pair of equations (from Baltic Way 2021) 
Show that no non-zero integers $a, b, x, y$ satisfy:
\begin{cases} ax-by=16. \\ ay+bx=1. \end{cases}

From Baltic-Way 2021.
\begin{align}
&(a+bi)(x+yi)=(ax-by)+i(ay+bx)=16+i. \\
&|(a+bi)(x+yi)|=|16+i|. \\
&\sqrt{a^2+b^2}\sqrt{x^2+y^2}=\sqrt{257}. \\
&(a^2+b^2)(x^2+y^2)=257. \\
&\therefore a^2+b^2=257, x^2+y^2=1 \text{ or } a^2+b^2=1, x^2+y^2=257.(\because 257: \text{ prime.)} \\
\Rightarrow & \text{No solution.}
\end{align}
Is my solution right?
 [/INST] Your solution is indeed valid, and as FShrike writes in the comments, quite elegant. I'm putting this confirmation in a community wiki answer to mark this question as answered. (I think this is how that works!)
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