From: Thom Mulders <mulders@inf.ethz.ch>
Subject: Re: no 5 integers satisfying ai*aj+1=square ?
Date: Thu, 30 Mar 2000 09:13:47 +0200
Newsgroups: sci.math.num-analysis

Pavel.Pokorny@vscht.cz wrote:

> How to prove that there are no (I assume) 5 integers satisfying
> ai*aj+1=square ?
> There are, however, 4 integers satisfying this. The least being
> 1,3,8,120 because
> 1*3+1 = 2*2 , 1*120+1 = 11*11
> 1*8+1 = 3*3 , 3*120+1 = 19*19
> 3*8+1 = 5*5 , 8*120+1 = 31*31
>
> --
> Pavel Pokorny
> Math Dept, Prague Institute of Chemical Technology
> http://staff.vscht.cz/mat/Pavel.Pokorny

A set of non-zero rational numbers {a1,...,ak} such that ai*aj+n=square
is called a D(n)-k-tuple.
I don't know if there are D(1)-quintuples but there is a
D(2^8)-quintuple, i.e. {1,33,105,320,18240}, found by Dujella.

At the Rhine Workshop on Computer Algebra, held on 22.3.00-24.3.00 in
Bergenz,  H.G. Zimmer of the
University of the Saarland in Saarbrucken (D), gave a talk on Fermat's
quadruple equations and he
used these to find the above quintuple. Maybe you can ask him if he
knows anything on
D(1)-quintuples.

I hope this helps,

Thom Mulders
ETH Zurich
Switzerland