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Imperfect Square Sum

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.

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

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