Total k-domination in Cartesian product graphs

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 detail, namely a problem that origins from a conjecture of Rényi and Erd?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). 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?r (Hungary).