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25x^2 - 144y^2 = 3600.
We seek integer solutions $(x, y)$. Rearranging:
25x^2 = 144y^2 + 3600.
We analyze modulo small values. First, note that $x^2 \geq 144$, so $|...
Let’s suppose $y$ is such that $144y^2 + 3600$ is divisible by 25 and...
Let $k^2 = \frac{144y^2 + 3600}{25}$. Then:
144y^2 + 3600 \equiv 0 \pmod{25} \Rightarrow 144y^2 \equiv -3600 \pmod...
Since $144 \equiv 19 \pmod{25}$ and $-3600 \equiv 0 \pmod{25}$, we get...
19y^2 \equiv 0 \pmod{25} \Rightarrow y^2 \equiv 0 \pmod{25} \Rightarro...
So let $y = 5k$. Substitute:

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