TITLE: Find number of Positive Integer Solution
QUESTION [5 upvotes]: $$2^a - 5^b7^c = 1$$
 So, $a,b,c>0$.
For solving this I observed that last digit of $ 2^a$ are among any of $2, 4, 8$ and $6$ and last digit of $5^b7^c$ is always $5$ . Therefor if  a  is only in the form of 4n this equation may be valid. now arbitrarily I put $a=8$ and $b=1$ and $c=2$  and I found $$2^8-5^17^2=11$$ this result gives a feelings thet there is no such solution of this problem. 
I want to know wheather I am in the appropriate track or I am wrong. 
Please help.

REPLY [3 votes]: Hint:since
$$2^a-1=5^b\cdot 7^c$$
if $b>0$ then we have
$$5|2^a-1\Longrightarrow 4|a\Longrightarrow 3|2^a-1$$
impossible