Add the numbers of any line (horizontal, vertical or diagonal) and the results are identical.
These examples show one solution, there are others.
 
3x3 (All lines total 15)

8 1 6
3 5 7
4 9 2

4x4 (all lines total 34)

 1 12  8 13
14  7 11  2
15  6 10  3
 4  9  5 16

5x5 (all lines total 65)

11 23  4 10 17
 2 14 20 21  8
25  7 13 19  1
18  5  6 12 24
 9 16 22  3 15

6x6 (all lines total 111)

11 18 13 29 36  4
16 14 12 34  5 30
15 10 17 33 28  8
20 27 22  2  9 31
25 23 21  7 32  3
24 19 26  6  1 35

7x7 (all lines total 175)

30 21 13  5 46 38 22
39 23 15 14  6 47 31
48 32 24 16  8  7 40
 1 41 33 25 17  9 49
10 43 42 34 26 18  2
19  3 44 36 35 27 11
28 12  4 45 37 29 20

8x8 (all lines total 260)

 1  9 24 40 32 48 49 57
 2 10 23 39 31 47 50 58
59 51 46 30 38 22 11  3
61 53 44 28 36 20 13  5
60 52 45 29 37 21 12  4
62 54 43 27 35 19 14  6
 7 15 18 34 26 42 55 63
 8 16 17 33 25 41 56 64