Prove that there is only one value fornthat make

2^8 + 2^11 + 2^n is a perfect square.

Here is my attempt :

2^8 + 2^11 + 2^n = m^2

2^n = m^2 - 2^8 - 2^11

2^n = (m-48)(m+48)

According to a theorem (I don't know what it is if somebody know please tell me) that there exist non-negative numbersandtso that m - 48 = 2^s , m + 48 = 2^t , s+t=n

So :

2^s + 48 = 2^t -48

2^s - 2^t = 96

2^s{2^(t-s) - 1} = 2^5 x 3

s = 5 and t = 7, so n= s+t = 12

Is right or there is a better way ? thanks