Il existe 4 périodes de 27 chiffres pour n/109 en base 9.

Pour toutes les fractions de n/109 en base 9, la période de 1/109 revient alors 27 fois (en orange)

1/109=0,006616540363860220517312128...

2/109=0,014334180737830441135624257...

3/109=0,022051731212800661654036386...

4/109=0,028668371576760882272348525...

5/109=0,036386022051731212800661654...

6/109=0,044113562425701433418073783...

7/109=0,051731212800661654036386022...

8/109=0,058447753264631874554708151...

9/109=0,066165403638602205173121280...

10/109=0,073783044113562425701433418...

11/109=0,081510584477532646318745547...

12/109=0,088227234852502866837157676...

13/109=0,105844775326463187455470815...

14/109=0,113562425701433418073783044...

15/109=0,121280066165403638602205173...

16/109=0,128006616540363860220517312...

17/109=0,135624257014334180737830441...

18/109=0,143341807378304411356242570...

19/109=0,151058447753264631874554708...

20/109=0,157676088227234852502866837...

21/109=0,165403638602205173121280066...

22/109=0,173121280066165403638602205...

23/109=0,180737830441135624257014334...

24/109=0,187455470815105844775326463...

25/109=0,205173121280066165403638602...

26/109=0,212800661654036386022051731...

27/109=0,220517312128006616540363860...

28/109=0,227234852502866837157676088...

29/109=0,234852502866837157676088227...

30/109=0,242570143341807378304411356...

31/109=0,250286683715767608822723485...

32/109=0,257014334180737830441135624...

33/109=0,264631874554708151058447753...

34/109=0,272348525028668371576760882...

35/109=0,280066165403638602205173121...

36/109=0,286683715767608822723485250...

37/109=0,304411356242570143341807378...

38/109=0,312128006616540363860220517...

39/109=0,318745547081510584477532646...

40/109=0,326463187455470815105844775...

41/109=0,334180737830441135624257014...

42/109=0,341807378304411356242570143...

43/109=0,348525028668371576760882272...

44/109=0,356242570143341807378304411...

45/109=0,363860220517312128006616540...

46/109=0,371576760882272348525028668...

47/109=0,378304411356242570143341807...

48/109=0,386022051731212800661654036...

49/109=0,403638602205173121280066165...

50/109=0,411356242570143341807378304...

51/109=0,418073783044113562425701433...

52/109=0,425701433418073783044113562...

53/109=0,433418073783044113562425701...

54/109=0,441135624257014334180737830...

55/109=0,447753264631874554708151058...

56/109=0,455470815105844775326463187...

57/109=0,463187455470815105844775326...

58/109=0,470815105844775326463187455...

59/109=0,477532646318745547081510584...

60/109=0,485250286683715767608822723...

61/109=0,502866837157676088227234852...

62/109=0,510584477532646318745547081...

63/109=0,517312128006616540363860220...

64/109=0,525028668371576760882272348...

65/109=0,532646318745547081510584477...

66/109=0,540363860220517312128006616...

67/109=0,547081510584477532646318745...

68/109=0,554708151058447753264631874...

69/109=0,562425701433418073783044113...

70/109=0,570143341807378304411356242...

71/109=0,576760882272348525028668371...

72/109=0,584477532646318745547081510...

73/109=0,602205173121280066165403638...

74/109=0,608822723485250286683715767...

75/109=0,616540363860220517312128006...

76/109=0,624257014334180737830441135...

77/109=0,631874554708151058447753264...

78/109=0,638602205173121280066165403...

79/109=0,646318745547081510584477532...

80/109=0,654036386022051731212800661...

81/109=0,661654036386022051731212800...

82/109=0,668371576760882272348525028...

83/109=0,676088227234852502866837157...

84/109=0,683715767608822723485250286...

85/109=0,701433418073783044113562425...

86/109=0,708151058447753264631874554...

87/109=0,715767608822723485250286683...

88/109=0,723485250286683715767608822...

89/109=0,731212800661654036386022051...

90/109=0,737830441135624257014334180...

91/109=0,745547081510584477532646318...

92/109=0,753264631874554708151058447...

93/109=0,760882272348525028668371576...

94/109=0,767608822723485250286683715...

95/109=0,775326463187455470815105844...

96/109=0,783044113562425701433418073...

97/109=0,800661654036386022051731212...

98/109=0,807378304411356242570143341...

99/109=0,815105844775326463187455470...

100/109=0,822723485250286683715767608...

101/109=0,830441135624257014334180737...

102/109=0,837157676088227234852502866...

103/109=0,844775326463187455470815105...

104/109=0,852502866837157676088227234...

105/109=0,860220517312128006616540363...

106/109=0,866837157676088227234852502...

107/109=0,874554708151058447753264631...

108/109=0,882272348525028668371576760...

On remarque que le produit du nombre de périodes (4) et de leurs longueurs (27) est égal à 108 et donc au premier -1.