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From: arvind@utcsri.UUCP
Newsgroups: ut.theory
Subject: THEORY NET: Primes between squares
Message-ID: <5423@utcsri.UUCP>
Date: Sun, 20-Sep-87 23:46:36 EDT
Article-I.D.: utcsri.5423
Posted: Sun Sep 20 23:46:36 1987
Date-Received: Mon, 21-Sep-87 02:37:01 EDT
Distribution: ut
Organization: CSRI, University of Toronto
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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