$(2^a -1)(2^b -1)=2^{2^c}+1$ has no nonnegative integer solutions

Question

$(2^a -1)(2^b -1)=2^{2^c}+1$ is not possible for a,b,c nonnegative integers.

Any solutions using parity

Approach: $(2^a -1)(2^b -1)=2^{2^c}+1\Rightarrow$

$2^{a+b}-2^a-2^b=2^{2^c}\Rightarrow$

Answer 1

Hint:

The binary expansion of natural numbers is unique. Assume there exists a solution and use that to obtain two distinct binary expansions of the same number thus yielding a contradiction.

Answer 2

Assume that there was a solution and wlog that $a<b$. Divide by $2^a$, then you have

$2^b-1-2^{b-a}=2^{2^c-a}$

The left side is clearly odd, the right side is even or a fraction if $2^c<a$. Therefore, there cannot exist such a solution.