Not Quite a Perfect Square

We can rewrite the eqn as2^a+7^c=13^b+182. Now for min both lhs and rhs should be as small as possible. Min value of b can be 2, in which case a=c=3. Now we may check for b=3 in case it has a smaller solution (it doesn't) and clearly bigger values of b will result in bigger values of a and c. Thus we can say it's a minimum value.

$2^3 = 13^2 + 182 - 7^3$ seems to be the only integral solution for which a + b + c is least. So $a = 3$ , $b = 2$ , $c = 3$

Now, as I found one of the solutions, so my second aim to check whether 8 is the smallest possible value. I also took advantage of the fact that neither a,b or c can be equal to 1. So a,b or c can have only 2 or 3 in its solution set. Which gives total 4 combinations $(2,2,2) , (2,2,3) ,(2,3,2), (3,2,2)$ where the sum of $a + b + c$ will be less than 8. But none of the 4 sets satisfy the given equation, so I can say $(3,2,3)$ is the only possible set where the value of $a + b + c$ is the least.

Thus $a + b + c = 3 + 2 + 3 = 8$