From: randall_rathbun@rc.trw.com
Date: Tue, 04 Aug 1998 20:35:27 -0800
To: rusin@math.niu.edu
Subject: Update to the Rational Box

Dave:
As of 1994, when I graduated from California State University
San Marcos, in May, I had searched for the rational box,
or integer cuboid, with the smallest side <= 1,281,000,000
with no success for a perfect cuboid. There were 30K+ solutions
of the 3 types of cuboids found (body,edge,face) but not even
any new information that John Leech was looking for regarding
face pentacycles was uncovered. Sigh. This is so you can update
your web page.
- Randall L. Rathbun
p.s: The search program was run twice, Torbjorn Granlund helped me,
using MP arithmetic, it was exhaustive over the range, and it
took 5 years computer time (IBM RS6000 workstations)
==============================================================================

From: Dave Rusin <rusin@math.niu.edu>
Date: Tue, 4 Aug 1998 22:51:22 -0500 (CDT)
To: randall_rathbun@rc.trw.com
Subject: Re:  Update to the Rational Box

Thanks. Do you do a more or less exhaustive search, or do you parameterize
the solution sets to partial configurations (e.g. boxes with integer sides
and a couple of integer diagonals) and then eliminate those? I ask because I
try such parameterizations from time to time and try "small" points of
those types; right this minute my machine is walking over a few elliptic curves
of rank 2 (which parameterize certain partial configurations) looking
for complete solutions with dozens of digits -- sort of a shot in the
dark, but hey, you never know!

dave
==============================================================================

Date: 5 Aug 1998 13:33:43 U
From: "Randall Rathbun" <Randall_Rathbun@smtp.rc.trw.com>
Subject: Re:  Update to the Rational
To: "Dave Rusin" <rusin@math.niu.edu>

                      RE>> Update to the Rational Box              8/5/98

Dave:
John Leech (when he was alive) and I did both kinds of searching. We did =
use
a parametric solution and looped over it. My exhaustive search was =
completely
exhaustive over all 3 types of cuboids, so there is NO possibility any =
more of
any perfect cuboid having any edge less than 1,281,000,000 . This is
discouraging and some other results of the pythagorean generators =
involved
seem to point that the perfect cuboid does not exist. (too many symmetry
requirements of the pythagorean generators)
- Randall