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?
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?
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.