## Imperfect Square Sum

#### Problem

Prove that `n`^{4} + 3`n`^{2} + 2 is never square.

#### Solution

By factoring `n`^{4} + 3`n`^{2} + 2 = (`n`^{2} + 1)(`n`^{2} + 2).

We can see that (`n`^{2} + 1)^{2} < (`n`^{2} + 1)(`n`^{2} + 2) < (`n`^{2} + 2)^{2}.

Hence `n`^{4} + 3`n`^{2} + 2 lies between two consecutive squares, so it cannot be a square itself.

Prove that `n`^{4} + 2`n`^{3} + 2`n`^{2} + 2`n` + 1 can never be square.

Problem ID: 133 (Nov 2003) Difficulty: 3 Star