Il existe 18 périodes de 7 chiffres pour n/127 en base 8.

Pour toutes les fractions de n/127 en base 8, la période de 1/127 revient alors 7 fois (en orange)

1/127=0,0040201...

2/127=0,0100402...

3/127=0,0140603...

4/127=0,0201004...

5/127=0,0241205...

6/127=0,0301406...

7/127=0,0341607...

8/127=0,0402010...

9/127=0,0442211...

10/127=0,0502412...

11/127=0,0542613...

12/127=0,0603014...

13/127=0,0643215...

14/127=0,0703416...

15/127=0,0743617...

16/127=0,1004020...

17/127=0,1044221...

18/127=0,1104422...

19/127=0,1144623...

20/127=0,1205024...

21/127=0,1245225...

22/127=0,1305426...

23/127=0,1345627...

24/127=0,1406030...

25/127=0,1446231...

26/127=0,1506432...

27/127=0,1546633...

28/127=0,1607034...

29/127=0,1647235...

30/127=0,1707436...

31/127=0,1747637...

32/127=0,2010040...

33/127=0,2050241...

34/127=0,2110442...

35/127=0,2150643...

36/127=0,2211044...

37/127=0,2251245...

38/127=0,2311446...

39/127=0,2351647...

40/127=0,2412050...

41/127=0,2452251...

42/127=0,2512452...

43/127=0,2552653...

44/127=0,2613054...

45/127=0,2653255...

46/127=0,2713456...

47/127=0,2753657...

48/127=0,3014060...

49/127=0,3054261...

50/127=0,3114462...

51/127=0,3154663...

52/127=0,3215064...

53/127=0,3255265...

54/127=0,3315466...

55/127=0,3355667...

56/127=0,3416070...

57/127=0,3456271...

58/127=0,3516472...

59/127=0,3556673...

60/127=0,3617074...

61/127=0,3657275...

62/127=0,3717476...

63/127=0,3757677...

64/127=0,4020100...

65/127=0,4060301...

66/127=0,4120502...

67/127=0,4160703...

68/127=0,4221104...

69/127=0,4261305...

70/127=0,4321506...

71/127=0,4361707...

72/127=0,4422110...

73/127=0,4462311...

74/127=0,4522512...

75/127=0,4562713...

76/127=0,4623114...

77/127=0,4663315...

78/127=0,4723516...

79/127=0,4763717...

80/127=0,5024120...

81/127=0,5064321...

82/127=0,5124522...

83/127=0,5164723...

84/127=0,5225124...

85/127=0,5265325...

86/127=0,5325526...

87/127=0,5365727...

88/127=0,5426130...

89/127=0,5466331...

90/127=0,5526532...

91/127=0,5566733...

92/127=0,5627134...

93/127=0,5667335...

94/127=0,5727536...

95/127=0,5767737...

96/127=0,6030140...

97/127=0,6070341...

98/127=0,6130542...

99/127=0,6170743...

100/127=0,6231144...

101/127=0,6271345...

102/127=0,6331546...

103/127=0,6371747...

104/127=0,6432150...

105/127=0,6472351...

106/127=0,6532552...

107/127=0,6572753...

108/127=0,6633154...

109/127=0,6673355...

110/127=0,6733556...

111/127=0,6773757...

112/127=0,7034160...

113/127=0,7074361...

114/127=0,7134562...

115/127=0,7174763...

116/127=0,7235164...

117/127=0,7275365...

118/127=0,7335566...

119/127=0,7375767...

120/127=0,7436170...

121/127=0,7476371...

122/127=0,7536572...

123/127=0,7576773...

124/127=0,7637174...

125/127=0,7677375...

126/127=0,7737576...

On remarque que le produit du nombre de périodes (18) et de leurs longueurs (7) est égal à 126 et donc au premier -1.