Have you tried induction?
Prove that there is only one value for n that 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 number s and t so 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