I am trying to solve the following exponential Diophantine equation:


$$ 9^{k\_1} -2^{j\_1} = 9^{k\_2}-2^{j\_2}$$


My conjecture is that this implies $k\_1=k\_2$ and $j\_1=j\_2$, apart from eventually some small few exceptions. Is this true?



Solution to an exponential Diophantine equation
Yes, this follows from a conjecture of Pillai (1945), which was proved by Stroeker and Tijdeman (1982). For references and generalizations see:


M. A. Bennett, Pillai’s conjecture revisited, J. Number Theory 98 (2003), 228-235.


R. Scott, R. Styer, On $p^x-q^y=c$ and related three term exponential Diophantine equations with prime bases, J. Number Theory 105 (2004), 212-234.


