Description
Online Number Theory Seminar
Abstract: Let $a,b,c$ be fixed coprime positive integers greater than 1, and let $N(a,b,c)$ denote the number of positive integer solutions $(x,y,z)$ of the equation $a^x+b^y = c^z.$ In the past hundred years, although there have been very rich results on the upper bound for $N(a,b,c),$ some key problems in this area still poorly understood. In 2016, R. Scott and R. Styer conjectured that $N(a,b,c)\leq 1,$ except with a few known exceptional cases. This is a difficult problem that is still far from being solved. In 2019, Y.-Z. Hu and M.-H. Le proved that if $\max\{a,b,c\}>10^{62},$ then $N(a,b,c)\leq 2.$ Afterwards, T. Miyazaki and I. Pink eliminated the condition $\max\{a,b,c\}>10^{62}$ from the above result. Now, we present the new advances of the above conjecture. Joint work with Reese Scott and Robert Styer.
For access please contact the organizers (ntrg[at]science.unideb.hu).