--- theo2357 wrote:

>

> Let p,q,r be three consecutive primes. How can be proven that

> 1/p < 1/q + 1/r? It's NOT trivial!

>

I think this is implied by the Iwaniec & Pintz finding, namely

that there is always a prime between x and x-x^(23/42), for

any real number x > 11.

The Iwaniec & Pintz theorem can be used to show the weaker result,

that for prime p >= 5, nextprime(p)/p < Phi, where Phi is the

golden ratio, (sqrt(5)+1)/2 = 1.618...

And if q/p < Phi, and r/q < Phi, then the original inequality

1/p < 1/q + 1/r follows quite easily.