TITLE: $ \frac{m^a}{2} + \frac{m}{2}-1 $ is not prime, $m \geq 4$ even integer, $ a\in \mathbb{Z}, a \geq 2 $
QUESTION [0 upvotes]: Question:
Prove that $ \frac{m^a}{2} + \frac{m}{2}-1 $ is not prime, $m \geq 4$ is an even integer, $ a\in \mathbb{Z}, a \geq 2 $.
Personal effort:
$m$ is an even, so $m=2k$.
$ \frac{(2k)^a}{2} + \frac{2k}{2}-1 =  2^{a-1}k^a + k -1 $
If $k$ is odd, then $ 2^{a-1}k^a + k -1 $ is even, bigger than $2$, so it can't be prime.
If $k$ is even (I didn't find anything useful yet)
Any help, ideas, hints, solutions? Thanks a lot

REPLY [1 votes]: Let $q=m^a/2+m/2-1$, then
$$2q\equiv m^a+m-2\equiv 0\pmod{m-1}.$$
Thus $m-1\mid 2q$. Since $m$ is even, $(2,m-1)=1$ and $m-1\mid q$. We know that $m-1\neq 1$ since $m\ge 4$. We also know that
$$m-1\neq m^a/2+m/2-1$$
because otherwise we'd have $m^{a-1}=1$ giving $m\le 1$. Hence $1<m-1<q$ making $q$ not prime.
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