30 years of collaboration

TL;DR

This paper reviews 30 years of collaboration between Austrian and Hungarian number theorists, highlighting key results and focusing on problems related to polynomial squares and the unit sum number problem.

Contribution

It provides a curated overview of significant results from a long-standing collaboration, emphasizing two specific open problems in number theory.

Findings

Discussion of a conjecture by Rényi and Erdős on polynomial squares

Analysis of Zelinsky's question on the unit sum number problem

Presentation of selected interesting data and results

Abstract

We highlight some of the most important cornerstones of the long standing and very fruitful collaboration of the Austrian Diophantine Number Theory research group and the Number Theory and Cryptography School of Debrecen. However, we do not plan to be complete in any sense but give some interesting data and selected results that we find particularly nice. At the end we focus on two topics in more details, namely a problem that origins from a conjecture of R\'enyi and Erd\H{o}s (on the number of terms of the square of a polynomial) and another one that origins from a question of Zelinsky (on the unit sum number problem), which will be presented in turn. This paper evolved from a plenary invited talk that the authors gave at the Joint Austrian-Hungarian Mathematical Conference 2015, August 25-27, 2015 in Gy\H{o}r (Hungary).