Prove that the equation $x^{3} - 3 x y^{2} + y^{3} = 2891$ has no solution in integers.

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Solution

Observe that $2891 = 7^{2} \times 59$ . If one of $x, y$ is divisible by $7$ , then so is the other one, and it follows that $7^{3}$ divides $2891$ , a contradiction. Since $7 | y$ , there exists $z$ such that $y z \equiv 1$ (mod $7$ ). Multiplying by $z^{3}$ and denoting $x z = t$ , we obtain $t^{3} - 3 t + 1 \equiv 0$ (mod $7$ ).

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