Proof, that the equation $$(x^2-13)(x^2-17)(x^2-13*17)$$ has no roots in $$\mathbb{Z}$$ but for every modulo $$m \in \mathbb{Z}_{\geq 2}$$. It is obvious that the equation has no solutions in $$\mathbb{Z}$$. I've already found out that the equation does have solutions modulo m for m=2,3,4.. and so on, but I am not sure how to prove it for every m. Maybe someone can give me a small hint :)
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It is obvious that the equation has no roots in $$\mathbb{Z}$$. I've already found out that the equation does have solutions modulo m for m=2, 3 and so on, but I have no idea how to prove it for every m. – lisanno Jan 12 '22 at 16:44
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1Can you settle it for all prime moduli? – lulu Jan 12 '22 at 16:45
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1Please read how to ask a good question. And [edit] your post accordingly. – ACB Jan 12 '22 at 16:46
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2Hint : What can you conclude if a prime is neither $13$ nor $17$ and neither a quadratic residue modulo $13$ nor modulo $17$ ? – Peter Jan 12 '22 at 17:04