Exponential diophantine equation

Question

Need some help regarding the equation $$2^a-3^b=(2^c-1)\cdot d >0$$ where $a,b,c,d$ are integers; $a,b$ are fixed; and $c>2$.

Can we show that $c,d$ exist? Thank you!

Answer 1

No. A counterexample is $2^6-3^3=37$, which is prime and not of the form $2^c-1$.