Il existe 2 périodes de 83 chiffres pour n/167 en base 4.

Pour toutes les fractions de n/167 en base 4, la période de 1/167 revient alors 83 fois (en orange)

1/167=0,00012020123211330022232300102121102301310200030100313023320111131200210302211203221...

2/167=0,00030100313023320111131200210302211203221000120201232113300222323001021211023013102...

3/167=0,00102121102301310200030100313023320111131200210302211203221000120201232113300222323...

4/167=0,00120201232113300222323001021211023013102000301003130233201111312002103022112032210...

5/167=0,00132222021331230311221301123332131321012200331110103323121223103202313330323302031...

6/167=0,00210302211203221000120201232113300222323001021211023013102000301003130233201111312...

7/167=0,00222323001021211023013102000301003130233201111312002103022112032210001202012321133...

8/167=0,00301003130233201111312002103022112032210001202012321133002223230010212110230131020...

9/167=0,00313023320111131200210302211203221000120201232113300222323001021211023013102000301...

10/167=0,00331110103323121223103202313330323302031001322220213312303112213011233321313210122...

11/167=0,01003130233201111312002103022112032210001202012321133002223230010212110230131020003...

12/167=0,01021211023013102000301003130233201111312002103022112032210001202012321133002223230...

13/167=0,01033231212231032023133303233020310013222202133123031122130112333213132101220033111...

14/167=0,01111312002103022112032210001202012321133002223230010212110230131020003010031302332...

15/167=0,01123332131321012200331110103323121223103202313330323302031001322220213312303112213...

16/167=0,01202012321133002223230010212110230131020003010031302332011113120021030221120322100...

17/167=0,01220033111010332312122310320231333032330203100132222021331230311221301123332131321...

18/167=0,01232113300222323001021211023013102000301003130233201111312002103022112032210001202...

19/167=0,01310200030100313023320111131200210302211203221000120201232113300222323001021211023...

20/167=0,01322220213312303112213011233321313210122003311101033231212231032023133303233020310...

21/167=0,02000301003130233201111312002103022112032210001202012321133002223230010212110230131...

22/167=0,02012321133002223230010212110230131020003010031302332011113120021030221120322100012...

23/167=0,02031001322220213312303112213011233321313210122003311101033231212231032023133303233...

24/167=0,02103022112032210001202012321133002223230010212110230131020003010031302332011113120...

25/167=0,02121102301310200030100313023320111131200210302211203221000120201232113300222323001...

26/167=0,02133123031122130112333213132101220033111010332312122310320231333032330203100132222...

27/167=0,02211203221000120201232113300222323001021211023013102000301003130233201111312002103...

28/167=0,02223230010212110230131020003010031302332011113120021030221120322100012020123211330...

29/167=0,02301310200030100313023320111131200210302211203221000120201232113300222323001021211...

30/167=0,02313330323302031001322220213312303112213011233321313210122003311101033231212231032...

31/167=0,02332011113120021030221120322100012020123211330022232300102121102301310200030100313...

32/167=0,03010031302332011113120021030221120322100012020123211330022232300102121102301310200...

33/167=0,03022112032210001202012321133002223230010212110230131020003010031302332011113120021...

34/167=0,03100132222021331230311221301123332131321012200331110103323121223103202313330323302...

35/167=0,03112213011233321313210122003311101033231212231032023133303233020310013222202133123...

36/167=0,03130233201111312002103022112032210001202012321133002223230010212110230131020003010...

37/167=0,03202313330323302031001322220213312303112213011233321313210122003311101033231212231...

38/167=0,03221000120201232113300222323001021211023013102000301003130233201111312002103022112...

39/167=0,03233020310013222202133123031122130112333213132101220033111010332312122310320231333...

40/167=0,03311101033231212231032023133303233020310013222202133123031122130112333213132101220...

41/167=0,03323121223103202313330323302031001322220213312303112213011233321313210122003311101...

42/167=0,10001202012321133002223230010212110230131020003010031302332011113120021030221120322...

43/167=0,10013222202133123031122130112333213132101220033111010332312122310320231333032330203...

44/167=0,10031302332011113120021030221120322100012020123211330022232300102121102301310200030...

45/167=0,10103323121223103202313330323302031001322220213312303112213011233321313210122003311...

46/167=0,10122003311101033231212231032023133303233020310013222202133123031122130112333213132...

47/167=0,10200030100313023320111131200210302211203221000120201232113300222323001021211023013...

48/167=0,10212110230131020003010031302332011113120021030221120322100012020123211330022232300...

49/167=0,10230131020003010031302332011113120021030221120322100012020123211330022232300102121...

50/167=0,10302211203221000120201232113300222323001021211023013102000301003130233201111312002...

51/167=0,10320231333032330203100132222021331230311221301123332131321012200331110103323121223...

52/167=0,10332312122310320231333032330203100132222021331230311221301123332131321012200331110...

53/167=0,11010332312122310320231333032330203100132222021331230311221301123332131321012200331...

54/167=0,11023013102000301003130233201111312002103022112032210001202012321133002223230010212...

55/167=0,11101033231212231032023133303233020310013222202133123031122130112333213132101220033...

56/167=0,11113120021030221120322100012020123211330022232300102121102301310200030100313023320...

57/167=0,11131200210302211203221000120201232113300222323001021211023013102000301003130233201...

58/167=0,11203221000120201232113300222323001021211023013102000301003130233201111312002103022...

59/167=0,11221301123332131321012200331110103323121223103202313330323302031001322220213312303...

60/167=0,11233321313210122003311101033231212231032023133303233020310013222202133123031122130...

61/167=0,11312002103022112032210001202012321133002223230010212110230131020003010031302332011...

62/167=0,11330022232300102121102301310200030100313023320111131200210302211203221000120201232...

63/167=0,12002103022112032210001202012321133002223230010212110230131020003010031302332011113...

64/167=0,12020123211330022232300102121102301310200030100313023320111131200210302211203221000...

65/167=0,12032210001202012321133002223230010212110230131020003010031302332011113120021030221...

66/167=0,12110230131020003010031302332011113120021030221120322100012020123211330022232300102...

67/167=0,12122310320231333032330203100132222021331230311221301123332131321012200331110103323...

68/167=0,12200331110103323121223103202313330323302031001322220213312303112213011233321313210...

69/167=0,12213011233321313210122003311101033231212231032023133303233020310013222202133123031...

70/167=0,12231032023133303233020310013222202133123031122130112333213132101220033111010332312...

71/167=0,12303112213011233321313210122003311101033231212231032023133303233020310013222202133...

72/167=0,12321133002223230010212110230131020003010031302332011113120021030221120322100012020...

73/167=0,12333213132101220033111010332312122310320231333032330203100132222021331230311221301...

74/167=0,13011233321313210122003311101033231212231032023133303233020310013222202133123031122...

75/167=0,13023320111131200210302211203221000120201232113300222323001021211023013102000301003...

76/167=0,13102000301003130233201111312002103022112032210001202012321133002223230010212110230...

77/167=0,13120021030221120322100012020123211330022232300102121102301310200030100313023320111...

78/167=0,13132101220033111010332312122310320231333032330203100132222021331230311221301123332...

79/167=0,13210122003311101033231212231032023133303233020310013222202133123031122130112333213...

80/167=0,13222202133123031122130112333213132101220033111010332312122310320231333032330203100...

81/167=0,13300222323001021211023013102000301003130233201111312002103022112032210001202012321...

82/167=0,13312303112213011233321313210122003311101033231212231032023133303233020310013222202...

83/167=0,13330323302031001322220213312303112213011233321313210122003311101033231212231032023...

84/167=0,20003010031302332011113120021030221120322100012020123211330022232300102121102301310...

85/167=0,20021030221120322100012020123211330022232300102121102301310200030100313023320111131...

86/167=0,20033111010332312122310320231333032330203100132222021331230311221301123332131321012...

87/167=0,20111131200210302211203221000120201232113300222323001021211023013102000301003130233...

88/167=0,20123211330022232300102121102301310200030100313023320111131200210302211203221000120...

89/167=0,20201232113300222323001021211023013102000301003130233201111312002103022112032210001...

90/167=0,20213312303112213011233321313210122003311101033231212231032023133303233020310013222...

91/167=0,20231333032330203100132222021331230311221301123332131321012200331110103323121223103...

92/167=0,20310013222202133123031122130112333213132101220033111010332312122310320231333032330...

93/167=0,20322100012020123211330022232300102121102301310200030100313023320111131200210302211...

94/167=0,21000120201232113300222323001021211023013102000301003130233201111312002103022112032...

95/167=0,21012200331110103323121223103202313330323302031001322220213312303112213011233321313...

96/167=0,21030221120322100012020123211330022232300102121102301310200030100313023320111131200...

97/167=0,21102301310200030100313023320111131200210302211203221000120201232113300222323001021...

98/167=0,21120322100012020123211330022232300102121102301310200030100313023320111131200210302...

99/167=0,21133002223230010212110230131020003010031302332011113120021030221120322100012020123...

100/167=0,21211023013102000301003130233201111312002103022112032210001202012321133002223230010...

101/167=0,21223103202313330323302031001322220213312303112213011233321313210122003311101033231...

102/167=0,21301123332131321012200331110103323121223103202313330323302031001322220213312303112...

103/167=0,21313210122003311101033231212231032023133303233020310013222202133123031122130112333...

104/167=0,21331230311221301123332131321012200331110103323121223103202313330323302031001322220...

105/167=0,22003311101033231212231032023133303233020310013222202133123031122130112333213132101...

106/167=0,22021331230311221301123332131321012200331110103323121223103202313330323302031001322...

107/167=0,22100012020123211330022232300102121102301310200030100313023320111131200210302211203...

108/167=0,22112032210001202012321133002223230010212110230131020003010031302332011113120021030...

109/167=0,22130112333213132101220033111010332312122310320231333032330203100132222021331230311...

110/167=0,22202133123031122130112333213132101220033111010332312122310320231333032330203100132...

111/167=0,22220213312303112213011233321313210122003311101033231212231032023133303233020310013...

112/167=0,22232300102121102301310200030100313023320111131200210302211203221000120201232113300...

113/167=0,22310320231333032330203100132222021331230311221301123332131321012200331110103323121...

114/167=0,22323001021211023013102000301003130233201111312002103022112032210001202012321133002...

115/167=0,23001021211023013102000301003130233201111312002103022112032210001202012321133002223...

116/167=0,23013102000301003130233201111312002103022112032210001202012321133002223230010212110...

117/167=0,23031122130112333213132101220033111010332312122310320231333032330203100132222021331...

118/167=0,23103202313330323302031001322220213312303112213011233321313210122003311101033231212...

119/167=0,23121223103202313330323302031001322220213312303112213011233321313210122003311101033...

120/167=0,23133303233020310013222202133123031122130112333213132101220033111010332312122310320...

121/167=0,23211330022232300102121102301310200030100313023320111131200210302211203221000120201...

122/167=0,23230010212110230131020003010031302332011113120021030221120322100012020123211330022...

123/167=0,23302031001322220213312303112213011233321313210122003311101033231212231032023133303...

124/167=0,23320111131200210302211203221000120201232113300222323001021211023013102000301003130...

125/167=0,23332131321012200331110103323121223103202313330323302031001322220213312303112213011...

126/167=0,30010212110230131020003010031302332011113120021030221120322100012020123211330022232...

127/167=0,30022232300102121102301310200030100313023320111131200210302211203221000120201232113...

128/167=0,30100313023320111131200210302211203221000120201232113300222323001021211023013102000...

129/167=0,30112333213132101220033111010332312122310320231333032330203100132222021331230311221...

130/167=0,30131020003010031302332011113120021030221120322100012020123211330022232300102121102...

131/167=0,30203100132222021331230311221301123332131321012200331110103323121223103202313330323...

132/167=0,30221120322100012020123211330022232300102121102301310200030100313023320111131200210...

133/167=0,30233201111312002103022112032210001202012321133002223230010212110230131020003010031...

134/167=0,30311221301123332131321012200331110103323121223103202313330323302031001322220213312...

135/167=0,30323302031001322220213312303112213011233321313210122003311101033231212231032023133...

136/167=0,31001322220213312303112213011233321313210122003311101033231212231032023133303233020...

137/167=0,31020003010031302332011113120021030221120322100012020123211330022232300102121102301...

138/167=0,31032023133303233020310013222202133123031122130112333213132101220033111010332312122...

139/167=0,31110103323121223103202313330323302031001322220213312303112213011233321313210122003...

140/167=0,31122130112333213132101220033111010332312122310320231333032330203100132222021331230...

141/167=0,31200210302211203221000120201232113300222323001021211023013102000301003130233201111...

142/167=0,31212231032023133303233020310013222202133123031122130112333213132101220033111010332...

143/167=0,31230311221301123332131321012200331110103323121223103202313330323302031001322220213...

144/167=0,31302332011113120021030221120322100012020123211330022232300102121102301310200030100...

145/167=0,31321012200331110103323121223103202313330323302031001322220213312303112213011233321...

146/167=0,31333032330203100132222021331230311221301123332131321012200331110103323121223103202...

147/167=0,32011113120021030221120322100012020123211330022232300102121102301310200030100313023...

148/167=0,32023133303233020310013222202133123031122130112333213132101220033111010332312122310...

149/167=0,32101220033111010332312122310320231333032330203100132222021331230311221301123332131...

150/167=0,32113300222323001021211023013102000301003130233201111312002103022112032210001202012...

151/167=0,32131321012200331110103323121223103202313330323302031001322220213312303112213011233...

152/167=0,32210001202012321133002223230010212110230131020003010031302332011113120021030221120...

153/167=0,32222021331230311221301123332131321012200331110103323121223103202313330323302031001...

154/167=0,32300102121102301310200030100313023320111131200210302211203221000120201232113300222...

155/167=0,32312122310320231333032330203100132222021331230311221301123332131321012200331110103...

156/167=0,32330203100132222021331230311221301123332131321012200331110103323121223103202313330...

157/167=0,33002223230010212110230131020003010031302332011113120021030221120322100012020123211...

158/167=0,33020310013222202133123031122130112333213132101220033111010332312122310320231333032...

159/167=0,33032330203100132222021331230311221301123332131321012200331110103323121223103202313...

160/167=0,33111010332312122310320231333032330203100132222021331230311221301123332131321012200...

161/167=0,33123031122130112333213132101220033111010332312122310320231333032330203100132222021...

162/167=0,33201111312002103022112032210001202012321133002223230010212110230131020003010031302...

163/167=0,33213132101220033111010332312122310320231333032330203100132222021331230311221301123...

164/167=0,33231212231032023133303233020310013222202133123031122130112333213132101220033111010...

165/167=0,33303233020310013222202133123031122130112333213132101220033111010332312122310320231...

166/167=0,33321313210122003311101033231212231032023133303233020310013222202133123031122130112...

On remarque que le produit du nombre de périodes (2) et de leurs longueurs (83) est égal à 166 et donc au premier -1.