arvind@utcsri.UUCP (09/21/87)
Date: Tue, 15 Sep 87 18:08:08 EDT From: Hugh_L._Montgomery@ub.cc.umich.edu Subject: primes between squares It is not known whether there is a prime between any two squares, although of course it is conjectured that there is. The best known upper bound on the gaps between primes is that there is a prime between x and x + x^(11/20 + epsilon). On RH it is known (Cramer?) that there is a prime between x and x + Cx^(1/2)log x. In the opposite direction it is known only that there are infinitely many gaps as long as (log p). (loglog p)(loglogloglog p)/(logloglog p)^2. This latter result is 40 years old (ignoring improvements in the numerical constant, which I omitted), and Erdos offers $10,000 for an improvement (something tending to infinity faster). I seems to recall that Erdos once offered 10^(10^ 10) $$'s for a proof of the assertion that there is a prime between consecutive squares. --Hugh L. Montgomery, University of Michigan