🚀 go-pugleaf

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