Il existe 6 périodes de 18 chiffres pour n/109 en base 4.

Pour toutes les fractions de n/109 en base 4, la période de 1/109 revient alors 18 fois (en orange)

1/109=0,000211210333122123...

2/109=0,001023021332310312...

3/109=0,001300232332033101...

4/109=0,002112103331221230...

5/109=0,002323320331010013...

6/109=0,003201131330132202...

7/109=0,010013002323320331...

8/109=0,010230213323103120...

9/109=0,011102030322231303...

10/109=0,011313301322020032...

11/109=0,012131112321202221...

12/109=0,013002323320331010...

13/109=0,013220200320113133...

14/109=0,020032011313301322...

15/109=0,020303222313030111...

16/109=0,021121033312212300...

17/109=0,021332310312001023...

18/109=0,022210121311123212...

19/109=0,023021332310312001...

20/109=0,023233203310100130...

21/109=0,030111020303222313...

22/109=0,030322231303011102...

23/109=0,031200102302133231...

24/109=0,032011313301322020...

25/109=0,032223130301110203...

26/109=0,033101001300232332...

27/109=0,033312212300021121...

28/109=0,100130023233203310...

29/109=0,101001300232332033...

30/109=0,101213111232120222...

31/109=0,102030322231303011...

32/109=0,102302133231031200...

33/109=0,103120010230213323...

34/109=0,103331221230002112...

35/109=0,110203032223130301...

36/109=0,111020303222313030...

37/109=0,111232120222101213...

38/109=0,112103331221230002...

39/109=0,112321202221012131...

40/109=0,113133013220200320...

41/109=0,120010230213323103...

42/109=0,120222101213111232...

43/109=0,121033312212300021...

44/109=0,121311123212022210...

45/109=0,122123000211210333...

46/109=0,123000211210333122...

47/109=0,123212022210121311...

48/109=0,130023233203310100...

49/109=0,130301110203032223...

50/109=0,131112321202221012...

51/109=0,131330132202003201...

52/109=0,132202003201131330...

53/109=0,133013220200320113...

54/109=0,133231031200102302...

55/109=0,200102302133231031...

56/109=0,200320113133013220...

57/109=0,201131330132202003...

58/109=0,202003201131330132...

59/109=0,202221012131112321...

60/109=0,203032223130301110...

61/109=0,203310100130023233...

62/109=0,210121311123212022...

63/109=0,210333122123000211...

64/109=0,211210333122123000...

65/109=0,212022210121311123...

66/109=0,212300021121033312...

67/109=0,213111232120222101...

68/109=0,213323103120010230...

69/109=0,220200320113133013...

70/109=0,221012131112321202...

71/109=0,221230002112103331...

72/109=0,222101213111232120...

73/109=0,222313030111020303...

74/109=0,223130301110203032...

75/109=0,230002112103331221...

76/109=0,230213323103120010...

77/109=0,231031200102302133...

78/109=0,231303011102030322...

79/109=0,232120222101213111...

80/109=0,232332033101001300...

81/109=0,233203310100130023...

82/109=0,300021121033312212...

83/109=0,300232332033101001...

84/109=0,301110203032223130...

85/109=0,301322020032011313...

86/109=0,302133231031200102...

87/109=0,303011102030322231...

88/109=0,303222313030111020...

89/109=0,310100130023233203...

90/109=0,310312001023021332...

91/109=0,311123212022210121...

92/109=0,312001023021332310...

93/109=0,312212300021121033...

94/109=0,313030111020303222...

95/109=0,313301322020032011...

96/109=0,320113133013220200...

97/109=0,320331010013002323...

98/109=0,321202221012131112...

99/109=0,322020032011313301...

100/109=0,322231303011102030...

101/109=0,323103120010230213...

102/109=0,323320331010013002...

103/109=0,330132202003201131...

104/109=0,331010013002323320...

105/109=0,331221230002112103...

106/109=0,332033101001300232...

107/109=0,332310312001023021...

108/109=0,333122123000211210...

On remarque que le produit du nombre de périodes (6) et de leurs longueurs (18) est égal à 108 et donc au premier -1.