proof that $\frac{m^a}{2}+\frac{m}{2}-1$ is not a prime number

Question

I'm having trouble proving the following:

If $m$ is an even number, $m\ge4$ and $a$ is an integer, $a\ge2$ then $$\frac{m^a}{2}+\frac{m}{2}-1$$ is not a prime number.

Usually, in this type of exercises, someone has to show that the number can be written as a product of more than two expressions but whatever I've tried doesn't work...Thank you!

Answer 1

$${m^a\over 2}+{m\over 2}-1 = {m^a+m-2\over 2} = {m^a-1 + m-1\over 2} $$

$$= {(m-1)(m^{a-1} + ...+m+1)+(m-1)\over 2} $$

$$= \underbrace{(m-1)}_{=a}\underbrace{m^{a-1} + ...+m+2\over 2}_{=b} $$

Clearly $a>1$ and $b>1$ and $b$ is an integer, since $m$ is even.

So, the number is product of two numbers both $>1$, so it is not a prime.

Answer 2

Hint:

$$m^a+m-2$$ is divisible by $m-1$ and is even as $m^a+m=m(1+m^{a-1})$ is even for integer $m$