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Question:

Determine all positive integers $x$ and $y$ that satisfy the equation $x^2 + 2 = 5y$.

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up vote 1 down vote accepted

Hint: When is $5y - 2$ a square? Modulo 5 this is $5y - 2 = 3$, but squares mod 5 are always 0, 1 or 4 so the equation has no solutions.

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HINT $\rm\ \ x^2 \equiv\ \ldots\ \:(mod\ 5)\ $ versus $\rm\ \mathbb Z^2\ \equiv\ \{0,\:\pm1,\:\pm2\}^2\ \equiv\ \ldots \ \:(mod\ 5)$

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