AoPS - Problem

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Problem

Source: Czech-Polish-Slovak Junior Match 2012, Team p3 CPSJ

Tags: Perfect Square, number theory

parmenides51

13.03.2020 00:58

Prove that if $n$ is a positive integer then $2 (n^2 + 1) - n$ is not a square of an integer.

lks010203

13.03.2020 16:14

Multiply $8$ on each side which leads to $(4n-1)^2+15=8m^2$ . By modulo 3, we can get that $4n-1$ and $m$ are both multiples of 3, which leads to contradiction on modulo 9.