An absolute bound for the size of Diophantine m-tuples
by
Andrej Dujella
Department of Mathematics, University of Zagreb

A set of m positive integers { a₁, a₂, ... , a_m} is called a Diophantine m-tuple if a_i·a_j+1 is a perfect square for all 1 <= i < j <= m. The first Diophantine quadruple, {1, 3, 8, 120}, was found by Fermat. A famous conjecture is that there does not exist a Diophantine quintuple. There is even a stronger version of this conjecture, namely that if we fix a Diophantine triple {a, b, c}, then there is unique positive integer d such that d > max{a, b, c} and {a, b, c, d} is a Diophantine quadruple.

We are able to prove this stronger conjecture for a large class of Diophantine triples, namely for triples satisfying some gap conditions like b > 4a, c > max{b¹³, 10²⁰} or c > max{b⁵, 10¹⁰²⁹}.

This result allows us to give an absolute bound for the size of Diophantine m-tuples. More precisely, we can prove that there does not exist a Diophantine 9-tuple and that there are only finitely many Diophantine 8-tuples.

http://www.math.hr/~duje/papers.html

Date received: March 1, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadz-16.