I'm working on some equations in number theory and I stuck on below inequality :
pα+1qβ+1−2pα+1qβ−2pαqβ+1+2pαqβ+pα+1+qβ+1−1>0
Here p and q are distinct prime numbers and p,q>2 and α,β 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 xy((p−2)(q−2)−2)+px+qy−1
where x=pa and y=qb. Since p,q are distinct odd positive integers, (p−2)(q−2)≥3.
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