I need to prove that a term of a certain form cannot be a perfect square.

**miatchguy** · October 8th 2010, 12:50 AM

This is part of a much larger problem.
I need to prove that:
$\forall n\in\mathbb{Z}, n^2 \neq (4k+3),\forall k\in\mathbb{Z}$

Please help me in any way you can. Thanks.

**aman_cc** · October 8th 2010, 01:38 AM

Hint - Take n = 4m + 1 or 4m+3 ? What do you see? You don't need to consider even 'n'

**Soroban** · October 8th 2010, 05:37 AM

Hello, miatchguy!

Prove that: . $\forall n\in\mathbb{Z},\: n^2 \neq (4k+3),\:\forall k\in\mathbb{Z}$

$\,n$ must be either even or odd: . $\begin{Bmatrix} n \:=\: 2m \\ \text{or} \\ n \:=\: 2m+1 \end{Bmatrix}$

If $\,n$ is even: . $n^2 \;=\;(2m)^2 \;=\;4m^2$
. . $\,n^2$ is a multiple of 4.

If $\,n$ is odd: . $n^2 \;=\;(2m+1)^2 \;=\;4m^2 + 4m + 1 \;=\;4(m^2+m)+1$
. . $\,n^2$ is one more than a multiple of 4.

Therefore, $\,n^2$ is never three more than a multiple of 4.

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