## 15. ÖMG-Kongress

Jahrestagung der Deutschen Mathematikervereinigung

#### 16. bis 22. September 2001 in Wien

**Sektion 2 - Zahlentheorie**

Donnerstag, 20. September 2001, 17.00, Hörsaal 41

**A Polynomial Variant of a Problem of Diophantus and Euler**
**Clemens Fuchs**, **TU Graz**
(Koautor: Andrej Dujella)

Let be an integer. A classical Diophantine -tuple with the proberty
is a set of positive integers such that the product of any two of
them increased by is a perfect square. There has been a lot of recent
activity in this subject area, most notably by the coauthor, and the main
problem of interest here is finding upper bounds on for which there can
exist Diophantine -tuples satisfying . In this paper, we study this
question in the ring
, and we prove that for there is no
Diophantine quadruple
of non-zero polynomials, not all four
constant, having the proberty . The main idea behind the proof is to
reduce the given problem, via a theory of Pell equations in
,
to a question about occurrences of common values in two binary recurrent
sequences of polynomials, a problem which is dealt with by looking at the
degrees and leading terms of the polynomials arising as members of these two
recurrent sequences.

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