Il existe 3 périodes de 22 chiffres pour n/67 en base 3.

Pour toutes les fractions de n/67 en base 3, la période de 1/67 revient alors 22 fois (en orange)

1/67=0,0001012122022212101002...

2/67=0,0002102021122201202011...

3/67=0,0010121220222121010020...

4/67=0,0011211120022110111022...

5/67=0,0020001012122022212101...

6/67=0,0021020211222012020110...

7/67=0,0022110111022001121112...

8/67=0,0100200010121220222121...

9/67=0,0101212202221210100200...

10/67=0,0110002102021122201202...

11/67=0,0111022001121112002211...

12/67=0,0112111200221101110220...

13/67=0,0120201100021020211222...

14/67=0,0121220222121010020001...

15/67=0,0200010121220222121010...

16/67=0,0201100021020211222012...

17/67=0,0202112220120201100021...

18/67=0,0210202112220120201100...

19/67=0,0211222012020110002102...

20/67=0,0220011211120022110111...

21/67=0,0221101110220011211120...

22/67=0,0222121010020001012122...

23/67=0,1000210202112220120201...

24/67=0,1002000101212202221210...

25/67=0,1010020001012122022212...

26/67=0,1011102200112111200221...

27/67=0,1012122022212101002000...

28/67=0,1020211222012020110002...

29/67=0,1022001121112002211011...

30/67=0,1100021020211222012020...

31/67=0,1101110220011211120022...

32/67=0,1102200112111200221101...

33/67=0,1110220011211120022110...

34/67=0,1112002211011102200112...

35/67=0,1120022110111022001121...

36/67=0,1121112002211011102200...

37/67=0,1122201202011000210202...

38/67=0,1200221101110220011211...

39/67=0,1202011000210202112220...

40/67=0,1210100200010121220222...

41/67=0,1211120022110111022001...

42/67=0,1212202221210100200010...

43/67=0,1220222121010020001012...

44/67=0,1222012020110002102021...

45/67=0,2000101212202221210100...

46/67=0,2001121112002211011102...

47/67=0,2002211011102200112111...

48/67=0,2011000210202112220120...

49/67=0,2012020110002102021122...

50/67=0,2020110002102021122201...

51/67=0,2021122201202011000210...

52/67=0,2022212101002000101212...

53/67=0,2101002000101212202221...

54/67=0,2102021122201202011000...

55/67=0,2110111022001121112002...

56/67=0,2111200221101110220011...

57/67=0,2112220120201100021020...

58/67=0,2121010020001012122022...

59/67=0,2122022212101002000101...

60/67=0,2200112111200221101110...

61/67=0,2201202011000210202112...

62/67=0,2202221210100200010121...

63/67=0,2211011102200112111200...

64/67=0,2212101002000101212202...

65/67=0,2220120201100021020211...

66/67=0,2221210100200010121220...

On remarque que le produit du nombre de périodes (3) et de leurs longueurs (22) est égal à 66 et donc au premier -1.