Il existe 18 périodes de 7 chiffres pour n/127 en base 4.

Pour toutes les fractions de n/127 en base 4, la période de 1/127 revient alors 7 fois (en orange)

1/127=0,0002001...

2/127=0,0010002...

3/127=0,0012003...

4/127=0,0020010...

5/127=0,0022011...

6/127=0,0030012...

7/127=0,0032013...

8/127=0,0100020...

9/127=0,0102021...

10/127=0,0110022...

11/127=0,0112023...

12/127=0,0120030...

13/127=0,0122031...

14/127=0,0130032...

15/127=0,0132033...

16/127=0,0200100...

17/127=0,0202101...

18/127=0,0210102...

19/127=0,0212103...

20/127=0,0220110...

21/127=0,0222111...

22/127=0,0230112...

23/127=0,0232113...

24/127=0,0300120...

25/127=0,0302121...

26/127=0,0310122...

27/127=0,0312123...

28/127=0,0320130...

29/127=0,0322131...

30/127=0,0330132...

31/127=0,0332133...

32/127=0,1000200...

33/127=0,1002201...

34/127=0,1010202...

35/127=0,1012203...

36/127=0,1020210...

37/127=0,1022211...

38/127=0,1030212...

39/127=0,1032213...

40/127=0,1100220...

41/127=0,1102221...

42/127=0,1110222...

43/127=0,1112223...

44/127=0,1120230...

45/127=0,1122231...

46/127=0,1130232...

47/127=0,1132233...

48/127=0,1200300...

49/127=0,1202301...

50/127=0,1210302...

51/127=0,1212303...

52/127=0,1220310...

53/127=0,1222311...

54/127=0,1230312...

55/127=0,1232313...

56/127=0,1300320...

57/127=0,1302321...

58/127=0,1310322...

59/127=0,1312323...

60/127=0,1320330...

61/127=0,1322331...

62/127=0,1330332...

63/127=0,1332333...

64/127=0,2001000...

65/127=0,2003001...

66/127=0,2011002...

67/127=0,2013003...

68/127=0,2021010...

69/127=0,2023011...

70/127=0,2031012...

71/127=0,2033013...

72/127=0,2101020...

73/127=0,2103021...

74/127=0,2111022...

75/127=0,2113023...

76/127=0,2121030...

77/127=0,2123031...

78/127=0,2131032...

79/127=0,2133033...

80/127=0,2201100...

81/127=0,2203101...

82/127=0,2211102...

83/127=0,2213103...

84/127=0,2221110...

85/127=0,2223111...

86/127=0,2231112...

87/127=0,2233113...

88/127=0,2301120...

89/127=0,2303121...

90/127=0,2311122...

91/127=0,2313123...

92/127=0,2321130...

93/127=0,2323131...

94/127=0,2331132...

95/127=0,2333133...

96/127=0,3001200...

97/127=0,3003201...

98/127=0,3011202...

99/127=0,3013203...

100/127=0,3021210...

101/127=0,3023211...

102/127=0,3031212...

103/127=0,3033213...

104/127=0,3101220...

105/127=0,3103221...

106/127=0,3111222...

107/127=0,3113223...

108/127=0,3121230...

109/127=0,3123231...

110/127=0,3131232...

111/127=0,3133233...

112/127=0,3201300...

113/127=0,3203301...

114/127=0,3211302...

115/127=0,3213303...

116/127=0,3221310...

117/127=0,3223311...

118/127=0,3231312...

119/127=0,3233313...

120/127=0,3301320...

121/127=0,3303321...

122/127=0,3311322...

123/127=0,3313323...

124/127=0,3321330...

125/127=0,3323331...

126/127=0,3331332...

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