The 17th power of 2 + the 12th power of 17 * 2 + the nth power of 2 is a complete square number. Find all positive integer values of n

The 17th power of 2 + the 12th power of 17 * 2 + the nth power of 2 is a complete square number. Find all positive integer values of n

Classified discussion
m^2=2^17+17*2^12+2^n=32*2^12+17*2^12+2^n=49*2^12+2^n.
① When n = 12,
m^2=2^12*【49+2^(n-12)】,
So 49 + 2 ^ (n-12) is a complete square number, let 7 ^ 2 + 2 ^ (n-12) = a ^ 2, then
(a+7)(a-7)=2^(n-12),
Obviously n = 12 does not hold
Because (a + 7, A-7) = (a + 7,14) = 1, 2, 7 or 14, (a + 7) (A-7) = 2 ^ (n-12) cannot be divisible by 7,
(a + 7, A-7) = 1 or 2
When (a + 7, A-7) = 1,
A-7 = 1, a + 7 = 2 ^ (n-12), that is 14 = 2 ^ (n-12) - 1, does not hold (n is not 12, fixed 2 ^ (12-n) is even)
When (a + 7, A-7) = 2,
A-7 = 2, a + 7 = 2 ^ (N-13), that is, 2 ^ (N-13) - 2 = 14, 2 ^ (N-13) = 16 = 2 ^ 4,
So N-13 = 4, that is, n = 17
In conclusion, n = 17