A note on the Diophantine equation $|a^x-b^y|=c$

Authors

  • Bo He
  • Alain Togbé
  • Shichun Yang

DOI:

https://doi.org/10.7146/math.scand.a-15149

Abstract

Let $a,b,$ and $c$ be positive integers. We show that if $(a,b) =(N^k-1,N)$, where $N,k\geq 2$, then there is at most one positive integer solution $(x,y)$ to the exponential Diophantine equation $|a^x-b^y|=c$, unless $(N,k)=(2,2)$. Combining this with results of Bennett [3] and the first author [6], we stated all cases for which the equation $|(N^k \pm 1)^x - N^y|=c$ has more than one positive integer solutions $(x,y)$.

Downloads

Published

2010-12-01

How to Cite

He, B., Togbé, A., & Yang, S. (2010). A note on the Diophantine equation $|a^x-b^y|=c$. MATHEMATICA SCANDINAVICA, 107(2), 161–173. https://doi.org/10.7146/math.scand.a-15149

Issue

Section

Articles