Results obtained based on Wilson’s theorem

Results obtained based on proposed theorem

n

mod n

k

1

2

3

1

2

3

4

2

1

0

2

1

1

0

3

2

0

3

2

1

0

4

6

3

5

24

0

5

6

4

0

6

120

1

7

720

0

7

30

3

0

8

5,040

1

9

40,320

1

10

362,880

1

11

3,628,800

0

11

210

10

0

12

39,916,800

1

13

479,001,600

0

13

2310

10

0

14

6,227,020,800

1

15

87,178,291,200

1

16

1,307,674,368,000

1

17

20,922,789,888,000

0

17

30,030

2

0

18

355,687,428,096,000

1

19

6,402,373,705,728,000

0

19

510,510

1

0

20

121,645,100,408,832,000

1

21

2,432,902,008,176,640,000

1

22

51,090,942,171,709,400,000

1

23

1,124,000,727,777,610,000,000

0

23

9,699,690

3

0

24

25,852,016,738,885,000,000,000

1

25

620,448,401,733,239,000,000,000

1

26

15,511,210,043,331,000,000,000,000

1

27

403,291,461,126,606,000,000,000,000

1

28

10,888,869,450,418,400,000,000,000,000

1

29

304,888,344,611,714,000,000,000,000,000

0

29

223,092,870

17

0

30

8,841,761,993,739,700,000,000,000,000,000

1

31

265,252,859,812,191,000,000,000,000,000,000

0

31

6,469,693,230

13

0

32

8,222,838,654,177,920,000,000,000,000,000,000

1

33

263,130,836,933,694,000,000,000,000,000,000,000

1

34

8,683,317,618,811,890,000,000,000,000,000,000,000

1

35

295,232,799,039,604,000,000,000,000,000,000,000,000

1

36

10,333,147,966,386,100,000,000,000,000,000,000,000,000

1

37

371,993,326,789,901,000,000,000,000,000,000,000,000,000

0

37

200,560,490,130

10

0

38

13,763,753,091,226,300,000,000,000,000,000,000,000,000,000

1

39

523,022,617,466,601,000,000,000,000,000,000,000,000,000,000

1

40

20,397,882,081,197,400,000,000,000,000,000,000,000,000,000,000

1

41

815,915,283,247,898,000,000,000,000,000,000,000,000,000,000,000

0

41

7,420,738,134,810

34

0