Thursday, 27 August 2015

elementary number theory - On the inequality palpha+1qbeta+12palpha+1qbeta2palphaqbeta+1+2palphaqbeta+palpha+1+qbeta+11>0



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



pα+1qβ+12pα+1qβ2pαqβ+1+2pαqβ+pα+1+qβ+11>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((p2)(q2)2)+px+qy1

where x=pa and y=qb. Since p,q are distinct odd positive integers, (p2)(q2)3.


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