Il existe 4 périodes de 27 chiffres pour n/109 en base 7.

Pour toutes les fractions de n/109 en base 7, la période de 1/109 revient alors 27 fois (en orange)

1/109=0,003101230402150651261414455...

2/109=0,006202461104331632553132243...

3/109=0,012304021506512614144550031...

4/109=0,015405252211663565436264516...

5/109=0,021506512614144550031012304...

6/109=0,024611043316325531322430062...

7/109=0,031012304021506512614144550...

8/109=0,034113534423660464205562335...

9/109=0,040215065126141445500310123...

10/109=0,043316325531322430062024611...

11/109=0,046420556233503411353442366...

12/109=0,052522116635654362645160154...

13/109=0,055623350341135344236604642...

14/109=0,062024611043316325531322430...

15/109=0,065126141445500310123040215...

16/109=0,101230402150651261414455003...

17/109=0,104331632553132243006202461...

18/109=0,110433163255313224300620246...

19/109=0,113534423660464205562335034...

20/109=0,116635654362645160154052522...

21/109=0,123040215065126141445500310...

22/109=0,126141445500310123040215065...

23/109=0,132243006202461104331632553...

24/109=0,135344236604642055623350341...

25/109=0,141445500310123040215065126...

26/109=0,144550031012304021506512614...

27/109=0,150651261414455003101230402...

28/109=0,154052522116635654362645160...

29/109=0,160154052522116635654362645...

30/109=0,163255313224300620246110433...

31/109=0,166356543626451601540525221...

32/109=0,202461104331632553132243006...

33/109=0,205562335034113534423660464...

34/109=0,211663565436264516015405252...

35/109=0,215065126141445500310123040...

36/109=0,221166356543626451601540525...

37/109=0,224300620246110433163255313...

38/109=0,230402150651261414455003101...

39/109=0,233503411353442366046420556...

40/109=0,236604642055623350341135344...

41/109=0,243006202461104331632553132...

42/109=0,246110433163255313224300620...

43/109=0,252211663565436264516015405...

44/109=0,255313224300620246110433163...

45/109=0,261414455003101230402150651...

46/109=0,264516015405252211663565436...

47/109=0,300620246110433163255313224...

48/109=0,304021506512614144550031012...

49/109=0,310123040215065126141445500...

50/109=0,313224300620246110433163255...

51/109=0,316325531322430062024611043...

52/109=0,322430062024611043316325531...

53/109=0,325531322430062024611043316...

54/109=0,331632553132243006202461104...

55/109=0,335034113534423660464205562...

56/109=0,341135344236604642055623350...

57/109=0,344236604642055623350341135...

58/109=0,350341135344236604642055623...

59/109=0,353442366046420556233503411...

60/109=0,356543626451601540525221166...

61/109=0,362645160154052522116635654...

62/109=0,366046420556233503411353442...

63/109=0,402150651261414455003101230...

64/109=0,405252211663565436264516015...

65/109=0,411353442366046420556233503...

66/109=0,414455003101230402150651261...

67/109=0,420556233503411353442366046...

68/109=0,423660464205562335034113534...

69/109=0,430062024611043316325531322...

70/109=0,433163255313224300620246110...

71/109=0,436264516015405252211663565...

72/109=0,442366046420556233503411353...

73/109=0,445500310123040215065126141...

74/109=0,451601540525221166356543626...

75/109=0,455003101230402150651261414...

76/109=0,461104331632553132243006202...

77/109=0,464205562335034113534423660...

78/109=0,500310123040215065126141445...

79/109=0,503411353442366046420556233...

80/109=0,506512614144550031012304021...

81/109=0,512614144550031012304021506...

82/109=0,516015405252211663565436264...

83/109=0,522116635654362645160154052...

84/109=0,525221166356543626451601540...

85/109=0,531322430062024611043316325...

86/109=0,534423660464205562335034113...

87/109=0,540525221166356543626451601...

88/109=0,543626451601540525221166356...

89/109=0,550031012304021506512614144...

90/109=0,553132243006202461104331632...

91/109=0,556233503411353442366046420...

92/109=0,562335034113534423660464205...

93/109=0,565436264516015405252211663...

94/109=0,601540525221166356543626451...

95/109=0,604642055623350341135344236...

96/109=0,611043316325531322430062024...

97/109=0,614144550031012304021506512...

98/109=0,620246110433163255313224300...

99/109=0,623350341135344236604642055...

100/109=0,626451601540525221166356543...

101/109=0,632553132243006202461104331...

102/109=0,635654362645160154052522116...

103/109=0,642055623350341135344236604...

104/109=0,645160154052522116635654362...

105/109=0,651261414455003101230402150...

106/109=0,654362645160154052522116635...

107/109=0,660464205562335034113534423...

108/109=0,663565436264516015405252211...

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