Examples of the multiplicative groups of integers mod p and sums of squares

See below.


p=13. Generator g=2. √(-1)=8

g0 1
g1 2
g2 4
g3 8
g4 3
g5 6
g6 12
g7 11
g8 9
g9 5
g10 10
g11 7


12+82 =65 =5×13=0 mod 13
22+32 =13 =0 mod 13
42+62 =52 =4×13=0 mod 13

122+82 =208 =16×13=0 mod 13
112+32 =130 =10×13=0 mod 13
92+62 =117 =9×13=0 mod 13

12+52 =26 =2×13=0 mod 13
22+102 =104 =8×13=0 mod 13
42+72 =65 =5×13=0 mod 13

122+52 =169 =13×13=0 mod 13
112+102 =221 =17×13=0 mod 13
92+72 =130 =10×13=0 mod 13


p=17. Generator g=3. √(-1)=13

g0 1
g1 3
g2 9
g3 10
g4 13
g5 5
g6 15
g7 11
g8 16
g9 14
g10 8
g11 7
g12 4
g13 12
g14 2
g15 6
g16 1


12+132 =170 =10×17=0 mod 17
32+52 =34 =2×17=0 mod 17
92+152 =306 =18×17=0 mod 17
102+112 =221 =13×17=0 mod 17

162+132 =425 =25×17=0 mod 17
142+52 =221 =13×17=0 mod 17
82+152 =289 =17×17=0 mod 17
72+112 =170 =10×17=0 mod 17

12+42 =17 =0 mod 17
32+122 =153 =9×17=0 mod 17
92+22 =85 =5×17=0 mod 17
102+62 =136 =8×17=0 mod 17

162+42 =271 =16×17=0 mod 17
142+122 =340 =20×17=0 mod 17
82+22 =68 =4×17=0 mod 17
72+62 =85 =5×17=0 mod 17