elementary number theory - On the inequality $p^{alpha +1}q^{beta +1}-2p^{alpha+1}q^{beta}-2p^{alpha}q^{beta +1}+2p^{alpha}q^{beta}+p^{alpha+1}+q^{beta+1}-1 > 0$

I'm working on some equations in number theory and I stuck on below inequality :

$$p^{\alpha +1}q^{\beta +1}-2p^{\alpha+1}q^{\beta}-2p^{\alpha}q^{\beta +1}+2p^{\alpha}q^{\beta}+p^{\alpha+1}+q^{\beta+1}-1 > 0$$

Here $p$ and $q$ are distinct prime numbers and $p,q >2$ and $\alpha\,,\beta$ are positive integer numbers.

Can somebody help me to prove that or find counterexample , although I believe that the inequality is true.

Answer

The LHS is $$x y (\;(p-2)(q-2)-2\;)+p x+q y-1$$ where $x=p^a$ and $y=q^b$. Since $p,q$ are distinct odd positive integers, $(p-2)(q-2)\geq 3.$