Find all pair $(x;y)$ of positive integers such that $k=\frac{2019(x^{4}+2)}{x^{2}y+1} \in \mathbb{N^*}$
$\Leftrightarrow 2019x^4-kyx^2+4038-k=0$.
Hence $\Delta\geqslant 0$
$\Leftrightarrow k^2y^2+4 \cdot 2019(k-4038)\geqslant 0\Leftrightarrow y^2\geqslant \frac{4 \cdot 2019(4038-k)}{k^2}$
$$x^2y+1 \mid 2019(x^2-2y)$$
I don't know what I should do next. Help me, please?
2026-05-04 21:57:50.1777931870
$\frac{2019(x^{4}+2)}{x^{2}y+1} \in \mathbb{N^*}$
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