5. Select some numbers without repetition from $1,2, \ldots 15$, such that the sum of any two numbers is not a perfect square of a natural number, then the maximum number of numbers that can be selected is ( ).
Answer: 8
Explanation: First, we argue that the first digit can just form the square number 16, with the groups being $(1,15) \quad(2,14) \ldots(7,9)$, (8)

There are 8 groups in total, and from each group, at most 1 number can be selected, so the maximum number of selections is 8.
Second, construction: if the maximum is 8, then there is no 1, there is 15, there is no 10, there is 6, there is no 3, there is 11, there is no 14, there is 2, there is no 7, there is 9, this is the only way to select 8 numbers.