Il existe 4 périodes de 34 chiffres pour n/137 en base 4.

Pour toutes les fractions de n/137 en base 4, la période de 1/137 revient alors 34 fois (en orange)

1/137=0,0001313211311232033320201220221013...

2/137=0,0003233023223130133301003101102032...

3/137=0,0011212301201022233221210321323111...

4/137=0,0013132113112320333202012202210130...

5/137=0,0021111331030213033122220023031203...

6/137=0,0023031203002111133103021303312222...

7/137=0,0031011020320003233023223130133301...

8/137=0,0032330232231301333010031011020320...

9/137=0,0100310110203200032330232231301333...

10/137=0,0102223322121032132311100112123012...

11/137=0,0110203200032330232231301333010031...

12/137=0,0112123012010222332212103213231110...

13/137=0,0120102223322121032132311100112123...

14/137=0,0122022101300013132113112320333202...

15/137=0,0130001313211311232033320201220221...

16/137=0,0131321131123203332020122022101300...

17/137=0,0133301003101102032000323302322313...

18/137=0,0201220221013000131321131123203332...

19/137=0,0203200032330232231301333010031011...

20/137=0,0211113310302130331222200230312030...

21/137=0,0213033122220023031203002111133103...

22/137=0,0221013000131321131123203332020122...

23/137=0,0222332212103213231110011212301201...

24/137=0,0230312030021111331030213033122220...

25/137=0,0232231301333010031011020320003233...

26/137=0,0300211113310302130331222200230312...

27/137=0,0302130331222200230312030021111331...

28/137=0,0310110203200032330232231301333010...

29/137=0,0312030021111331030213033122220023...

30/137=0,0320003233023223130133301003101102...

31/137=0,0321323111001121230120102223322121...

32/137=0,0323302322313013330100310110203200...

33/137=0,0331222200230312030021111331030213...

34/137=0,0333202012202210130001313211311232...

35/137=0,1001121230120102223322121032132311...

36/137=0,1003101102032000323302322313013330...

37/137=0,1011020320003233023223130133301003...

38/137=0,1013000131321131123203332020122022...

39/137=0,1020320003233023223130133301003101...

40/137=0,1022233221210321323111001121230120...

41/137=0,1030213033122220023031203002111133...

42/137=0,1032132311100112123012010222332212...

43/137=0,1100112123012010222332212103213231...

44/137=0,1102032000323302322313013330100310...

45/137=0,1110011212301201022233221210321323...

46/137=0,1111331030213033122220023031203002...

47/137=0,1113310302130331222200230312030021...

48/137=0,1121230120102223322121032132311100...

49/137=0,1123203332020122022101300013132113...

50/137=0,1131123203332020122022101300013132...

51/137=0,1133103021303312222002303120300211...

52/137=0,1201022233221210321323111001121230...

53/137=0,1203002111133103021303312222002303...

54/137=0,1210321323111001121230120102223322...

55/137=0,1212301201022233221210321323111001...

56/137=0,1220221013000131321131123203332020...

57/137=0,1222200230312030021111331030213033...

58/137=0,1230120102223322121032132311100112...

59/137=0,1232033320201220221013000131321131...

60/137=0,1300013132113112320333202012202210...

61/137=0,1301333010031011020320003233023223...

62/137=0,1303312222002303120300211113310302...

63/137=0,1311232033320201220221013000131321...

64/137=0,1313211311232033320201220221013000...

65/137=0,1321131123203332020122022101300013...

66/137=0,1323111001121230120102223322121032...

67/137=0,1331030213033122220023031203002111...

68/137=0,1333010031011020320003233023223130...

69/137=0,2000323302322313013330100310110203...

70/137=0,2002303120300211113310302130331222...

71/137=0,2010222332212103213231110011212301...

72/137=0,2012202210130001313211311232033320...

73/137=0,2020122022101300013132113112320333...

74/137=0,2022101300013132113112320333202012...

75/137=0,2030021111331030213033122220023031...

76/137=0,2032000323302322313013330100310110...

77/137=0,2033320201220221013000131321131123...

78/137=0,2101300013132113112320333202012202...

79/137=0,2103213231110011212301201022233221...

80/137=0,2111133103021303312222002303120300...

81/137=0,2113112320333202012202210130001313...

82/137=0,2121032132311100112123012010222332...

83/137=0,2123012010222332212103213231110011...

84/137=0,2130331222200230312030021111331030...

85/137=0,2132311100112123012010222332212103...

86/137=0,2200230312030021111331030213033122...

87/137=0,2202210130001313211311232033320201...

88/137=0,2210130001313211311232033320201220...

89/137=0,2212103213231110011212301201022233...

90/137=0,2220023031203002111133103021303312...

91/137=0,2222002303120300211113310302130331...

92/137=0,2223322121032132311100112123012010...

93/137=0,2231301333010031011020320003233023...

94/137=0,2233221210321323111001121230120102...

95/137=0,2301201022233221210321323111001121...

96/137=0,2303120300211113310302130331222200...

97/137=0,2311100112123012010222332212103213...

98/137=0,2313013330100310110203200032330232...

99/137=0,2320333202012202210130001313211311...

100/137=0,2322313013330100310110203200032330...

101/137=0,2330232231301333010031011020320003...

102/137=0,2332212103213231110011212301201022...

103/137=0,3000131321131123203332020122022101...

104/137=0,3002111133103021303312222002303120...

105/137=0,3010031011020320003233023223130133...

106/137=0,3012010222332212103213231110011212...

107/137=0,3013330100310110203200032330232231...

108/137=0,3021303312222002303120300211113310...

109/137=0,3023223130133301003101102032000323...

110/137=0,3031203002111133103021303312222002...

111/137=0,3033122220023031203002111133103021...

112/137=0,3101102032000323302322313013330100...

113/137=0,3103021303312222002303120300211113...

114/137=0,3111001121230120102223322121032132...

115/137=0,3112320333202012202210130001313211...

116/137=0,3120300211113310302130331222200230...

117/137=0,3122220023031203002111133103021303...

118/137=0,3130133301003101102032000323302322...

119/137=0,3132113112320333202012202210130001...

120/137=0,3200032330232231301333010031011020...

121/137=0,3202012202210130001313211311232033...

122/137=0,3203332020122022101300013132113112...

123/137=0,3211311232033320201220221013000131...

124/137=0,3213231110011212301201022233221210...

125/137=0,3221210321323111001121230120102223...

126/137=0,3223130133301003101102032000323302...

127/137=0,3231110011212301201022233221210321...

128/137=0,3233023223130133301003101102032000...

129/137=0,3301003101102032000323302322313013...

130/137=0,3302322313013330100310110203200032...

131/137=0,3310302130331222200230312030021111...

132/137=0,3312222002303120300211113310302130...

133/137=0,3320201220221013000131321131123203...

134/137=0,3322121032132311100112123012010222...

135/137=0,3330100310110203200032330232231301...

136/137=0,3332020122022101300013132113112320...

On remarque que le produit du nombre de périodes (4) et de leurs longueurs (34) est égal à 136 et donc au premier -1.