A note on the Diophantine equation $|a^x-b^y|=c$ - He, Bo; Togbé, Alain; Yang, Shichun (Mathematica Scandinavica, 2010)
Let $a,b,$ and $c$ be positive integers. We show that if $(a,b) =(N^k-1,N)$, where $N,k≥ 2$, 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 $|(N^k ± 1)^x - N^y|=c$ has more than one positive integer solutions $(x,y)$.