Hi How can be proved by using chebyshev's theorem(?, which theorem?) that for consecutive primes we have
(1+1/p^2)(1+1/q^2)<(1+1/r)
where p,q,r are consecutive primes and greater than 11
thanks
Hi How can be proved by using chebyshev's theorem(?, which theorem?) that for consecutive primes we have (1+1/p^2)(1+1/q^2)<(1+1/r) where p,q,r are consecutive primes and greater than 11 thanks 


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Replace $q=p$, the LHS gets bigger. Replace $r=4p$, the RHS gets smaller by the Chebyshev theorem (see Wiki). Now the inequality becomes, after simplification, $8p^2+4p^3<0$ which is true for $p\ge 11$. 

