Imperfect Square Sum
Prove that n4 + 3n2 + 2 is never square.
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.