Is this a square number?

Algebra Level 2

Given $A=n^4+2n^3+2n^2+2n+1$ with $n$ is a positive integer.

If $A=a^2$ , where $a$ is a positive integer, type $a$ , otherwise type $-1$ .

The answer is -1.

1 solution

Chew-Seong Cheong
Oct 20, 2019

$\begin{aligned} A & = n^4+2n^3 + 2n^2+2n+1 \\ & = \left(n^4+2n^2+1 \right) + 2n^3+2n \\ & = \left(n^2+1 \right)^2 + 2n\left(n^2+1 \right) \\ & = \left(n^2+1 \right)\left(n^2+2n+1 \right) \\ & = \left(n^2+1 \right)\left(n+1 \right)^2 \end{aligned}$

$A$ is a perfect square only if $n^2+1$ is a perfect square. But $n^2 + 1$ is not a perfect square for all positive integer $n$ . Therefore, the answer is $\boxed{-1}$ .

Is this a square number?

The answer is -1.

1 solution

1 pending report

Vote up reports you agree with