$(2^a -1)(2^b -1)=2^{2^c}+1$ has no nonnegative integer solutions $(2^a -1)(2^b -1)=2^{2^c}+1$ is not possible for a,b,c nonnegative integers.
Any solutions using parity
Approach:
$(2^a -1)(2^b -1)=2^{2^c}+1\Rightarrow$
$2^{a+b}-2^a-2^b=2^{2^c}\Rightarrow$

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.
