Question: Find all integer solutions of $y^2 = x^5 - 15x^4 + 10x^2 + 4x + 3$, or prove that none exist
Can anyone please solve this question for me? I have no idea how to start, so any comments are welcome!
Thanks
2026-03-31 21:56:33.1774994193
Find all integer solutions of $y^2 = x^5 - 15x^4 + 10x^2 + 4x + 3$, or prove that none exist
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If we reduce modulo $5$ (as suggested in comments) we get $$y^2\equiv x^5+4x+3\equiv x^5-x+3$$ By Fermat's little theorem, $x^5-x\equiv0\bmod5$. Thus $y^2\equiv3\bmod5$, which is impossible; $3$ is not a quadratic residue modulo $5$. Hence the original equation has no integral solutions.