mathschallenge.net

Problem

Prove that n⁴ + 3n² + 2 is never square.

Solution

By factoring n⁴ + 3n² + 2 = (n² + 1)(n² + 2).

We can see that (n² + 1)² < (n² + 1)(n² + 2) < (n² + 2)².

Hence n⁴ + 3n² + 2 lies between two consecutive squares, so it cannot be a square itself.

Prove that n⁴ + 2n³ + 2n² + 2n + 1 can never be square.