2022/061) What are the values of n for which $2^4+2^7+2^n$ is a perfect square?

 We have $2^4+2^7 + 2^n = 2^n + 144 = m^2$ where m is positive 

or $2^n = m^2 - 144 = (m+12)(m-12)$

as 2 is a prime so both m+ 12 and m-12 powers of 2

now difference (m+12)- (m-12) = 24

so powers of must have a difference 24

as $2^5 = 32$ so $2^6-2^k \ge 2^6 - 2^5 > 32$ (for any k less of equals 5)

so we need to check for candidates 2,4,8,16,32 and get 8 and 32.

this gives m = 20 and putting m = 20 we get n= 8