Course: Foundations of Higher Mathematics

Prove that $\displaystyle 2|(n^4 - 3)$ if and only if $\displaystyle 4|(n^2+3)$

I feel confident about this one. It's only the first part of the biconditional. Here's my attempt.

Assume that $\displaystyle 2|(n^4-3)$ , i.e. $\displaystyle n^4-3=2a$, for some integer a.

Then, $\displaystyle n^4=2a+3$

$\displaystyle =2(a+1) +1$

$\displaystyle =2b+1$.

Since $\displaystyle a+1$ is an integer, b is an integer also.

Therefore $\displaystyle n^4$ is odd, which means that $\displaystyle n$ is odd, i.e. $\displaystyle n=2c+1$, for some integer c.

Then, $\displaystyle n^2+3=(2c+1)^2+3$

$\displaystyle =4c^2+4c+4$

$\displaystyle =4(c^2+c+1)=4d$, for some integer d

Therefore $\displaystyle n^2+3=4d$, hence $\displaystyle 4|(n^2+3)$