Chebyshev theorem, consecutive primes - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-24T13:38:34Z http://mathoverflow.net/feeds/question/47668 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/47668/chebyshev-theorem-consecutive-primes Chebyshev theorem, consecutive primes asad 2010-11-29T12:15:40Z 2010-11-29T12:34:31Z <p>Hi How can be proved by using chebyshev's theorem(?, which theorem?) that for consecutive primes we have</p> <p>(1+1/p^2)(1+1/q^2)&lt;(1+1/r)</p> <p>where p,q,r are consecutive primes and greater than 11</p> <p>thanks</p> http://mathoverflow.net/questions/47668/chebyshev-theorem-consecutive-primes/47669#47669 Answer by Mark Sapir for Chebyshev theorem, consecutive primes Mark Sapir 2010-11-29T12:27:58Z 2010-11-29T12:34:31Z <p>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+4-p^3&lt;0\$ which is true for \$p\ge 11\$. </p>