Explanation

n^2 + 48 = m^2 gives (m−n)(m+n) = 48 with both factors of the same parity. Checking n < 5: n = 1 gives 49 = 7^2 and n = 4 gives 64 = 8^2, while n = 2, 3 give 52, 57 (not squares). Two such n exist.