Description
Online Number Theory Seminar
Abstract: After a short survey on well-known results on Pillai's equation $a^x-b^y = c$ we present extensions involving linear recurrences $(U_n).$ In particular, we show under mild technical conditions that the equation $U_n − b^m = c$ has at most 2 solutions $(n, m)$ for $b > B, n > N_0$ (effectively computable, joint work with Heintze, Vukusic and Ziegler). Furthermore, we consider Pillai's density problem, obtaining asymptotic results for the number of $c \in [1, x]$ such that $c = U_n- V_m$ for some $n, m \in\mathbb{N},$ (joint work with Vukusic, Yang and Ziegler). Other results are devoted to the so-called Pillai-Tijdeman equation. In the final section of the talk a new application of diophantine number theory to purely algebraic problems is discussed. It is shown that the binomial polynomials are absolutely irreducible in the ring Int $\mathbb{Z}$ of integer-valued polynomials over $\mathbb{Q}.$ This result is due to Rissner and Windisch and involves linear algebra tools as well as results on Pillai type equations and elementary prime number theory.
For access please contact the organizers (ntrg[at]science.unideb.hu).