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 $ n$ be an integer. A classical Diophantine $ m$-tuple with the proberty $ D(n)$ is a set of $ m$ positive integers such that the product of any two of them increased by $ n$ 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 $ m$ for which there can exist Diophantine $ m$-tuples satisfying $ D(n)$. In this paper, we study this question in the ring $ {\mathbb{Z}}[x]$, and we prove that for $ n=-1$ there is no Diophantine quadruple $ \{a,~b,~c,~d\}$ of non-zero polynomials, not all four constant, having the proberty $ D(-1)$. The main idea behind the proof is to reduce the given problem, via a theory of Pell equations in $ {\mathbb{Z}}[x]$, 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.

E-Mail: clemens.fuchs@tugraz.at
Homepage: finanz.math.tu-graz.ac.at/~fuchs/

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