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Originally Posted by Crook
Let P(n) denote the n-th prime number. Then, does anybody have an idea why P(n+1)<(sqrt(P(n))+1)^2 is true? This would be a lower bound than the Tchebycheff result that there is always a prime between n and 2n. Regards.
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The short answer is no. Noone does. We have no proof that it is
true. The best that has been achieved, when last I looked was that
there is always a prime between x and x + x^(11/20 + epsilon), for epsilon
depending on x as x -->oo. The fraction 11/20 may have been improved.
Note that even R.H does not yield the result you want. R.H. would imply
there is always a prime between x and x + sqrt(x)log x for sufficiently
large x. You want one between x and 2sqrt(x)+1.