Philadelphia Area Number Theory Seminar
Wednesday, February 15, 2023 - 3:30pm
Ingrid Vukusic
University of Salzburg and Franklin & Marshall College
An n-tuple of integers (a1; : : : ; an) is called multiplicatively dependent, if it allows you to win the \Cancelling Game", i.e. if there exist integers k1; : : : ; kn 2 Z, not all zero, such that ak1 1 akn n = 1: After an unconventional introduction, we will ask many questions related to consec- utive tuples of multiplicatively dependent integers, and answer some of them. For example, do there exist integers 1 < a < b such that (a; b) and (a + 1; b + 1) are both multiplicatively dependent? It turns out that this question is easily answered, and after briefly discussing some more general properties of pairs, we will move on to triples. The proof of the main result relies on lower bounds for linear forms in logarithms. This talk is based on joint work with Volker Ziegler, as well as some work in progress.