Il existe 2 périodes de 83 chiffres pour n/167 en base 3.

Pour toutes les fractions de n/167 en base 3, la période de 1/167 revient alors 83 fois (en orange)

1/167=0,00001110021202112101111202001021220220020210100012211011001010120000222012011200121...

2/167=0,00002220120112001210000111002120211210111120200102122022002021010001221101100101012...

3/167=0,00011100212021121011112020010212202200202101000122110110010101200002220120112001210...

4/167=0,00012211011001010120000222012011200121000011100212021121011112020010212202200202101...

5/167=0,00021021102210122221112201020110121111020221201002002202012122210011211221212102222...

6/167=0,00022201201120012100001110021202112101111202001021220220020210100012211011001010120...

7/167=0,00101012000022201201120012100001110021202112101111202001021220220020210100012211011...

8/167=0,00102122022002021010001221101100101012000022201201120012100001110021202112101111202...

9/167=0,00111002120211210111120200102122022002021010001221101100101012000022201201120012100...

10/167=0,00112112212121022220002102110221012222111220102011012111102022120100200220201212221...

11/167=0,00121000011100212021121011112020010212202200202101000122110110010101200002220120112...

12/167=0,00122110110010101200002220120112001210000111002120211210111120200102122022002021010...

13/167=0,00200220201212221001121122121210222200021021102210122221112201020110121111020221201...

14/167=0,00202101000122110110010101200002220120112001210000111002120211210111120200102122022...

15/167=0,00210211022101222211122010201101211110202212010020022020121222100112112212121022220...

16/167=0,00212021121011112020010212202200202101000122110110010101200002220120112001210000111...

17/167=0,00220201212221001121122121210222200021021102210122221112201020110121111020221201002...

18/167=0,00222012011200121000011100212021121011112020010212202200202101000122110110010101200...

19/167=0,01000122110110010101200002220120112001210000111002120211210111120200102122022002021...

20/167=0,01002002202012122210011211221212102222000210211022101222211122010201101211110202212...

21/167=0,01010120000222012011200121000011100212021121011112020010212202200202101000122110110...

22/167=0,01012000022201201120012100001110021202112101111202001021220220020210100012211011001...

23/167=0,01020110121111020221201002002202012122210011211221212102222000210211022101222211122...

24/167=0,01021220220020210100012211011001010120000222012011200121000011100212021121011112020...

25/167=0,01100101012000022201201120012100001110021202112101111202001021220220020210100012211...

26/167=0,01101211110202212010020022020121222100112112212121022220002102110221012222111220102...

27/167=0,01110021202112101111202001021220220020210100012211011001010120000222012011200121000...

28/167=0,01111202001021220220020210100012211011001010120000222012011200121000011100212021121...

29/167=0,01120012100001110021202112101111202001021220220020210100012211011001010120000222012...

30/167=0,01121122121210222200021021102210122221112201020110121111020221201002002202012122210...

31/167=0,01200002220120112001210000111002120211210111120200102122022002021010001221101100101...

32/167=0,01201120012100001110021202112101111202001021220220020210100012211011001010120000222...

33/167=0,01210000111002120211210111120200102122022002021010001221101100101012000022201201120...

34/167=0,01211110202212010020022020121222100112112212121022220002102110221012222111220102011...

35/167=0,01212221001121122121210222200021021102210122221112201020110121111020221201002002202...

36/167=0,01221101100101012000022201201120012100001110021202112101111202001021220220020210100...

37/167=0,01222211122010201101211110202212010020022020121222100112112212121022220002102110221...

38/167=0,02001021220220020210100012211011001010120000222012011200121000011100212021121011112...

39/167=0,02002202012122210011211221212102222000210211022101222211122010201101211110202212010...

40/167=0,02011012111102022120100200220201212221001121122121210222200021021102210122221112201...

41/167=0,02012122210011211221212102222000210211022101222211122010201101211110202212010020022...

42/167=0,02021010001221101100101012000022201201120012100001110021202112101111202001021220220...

43/167=0,02022120100200220201212221001121122121210222200021021102210122221112201020110121111...

44/167=0,02101000122110110010101200002220120112001210000111002120211210111120200102122022002...

45/167=0,02102110221012222111220102011012111102022120100200220201212221001121122121210222200...

46/167=0,02110221012222111220102011012111102022120100200220201212221001121122121210222200021...

47/167=0,02112101111202001021220220020210100012211011001010120000222012011200121000011100212...

48/167=0,02120211210111120200102122022002021010001221101100101012000022201201120012100001110...

49/167=0,02122022002021010001221101100101012000022201201120012100001110021202112101111202001...

50/167=0,02200202101000122110110010101200002220120112001210000111002120211210111120200102122...

51/167=0,02202012122210011211221212102222000210211022101222211122010201101211110202212010020...

52/167=0,02210122221112201020110121111020221201002002202012122210011211221212102222000210211...

53/167=0,02212010020022020121222100112112212121022220002102110221012222111220102011012111102...

54/167=0,02220120112001210000111002120211210111120200102122022002021010001221101100101012000...

55/167=0,02222000210211022101222211122010201101211110202212010020022020121222100112112212121...

56/167=0,10000111002120211210111120200102122022002021010001221101100101012000022201201120012...

57/167=0,10001221101100101012000022201201120012100001110021202112101111202001021220220020210...

58/167=0,10010101200002220120112001210000111002120211210111120200102122022002021010001221101...

59/167=0,10011211221212102222000210211022101222211122010201101211110202212010020022020121222...

60/167=0,10020022020121222100112112212121022220002102110221012222111220102011012111102022120...

61/167=0,10021202112101111202001021220220020210100012211011001010120000222012011200121000011...

62/167=0,10100012211011001010120000222012011200121000011100212021121011112020010212202200202...

63/167=0,10101200002220120112001210000111002120211210111120200102122022002021010001221101100...

64/167=0,10110010101200002220120112001210000111002120211210111120200102122022002021010001221...

65/167=0,10111120200102122022002021010001221101100101012000022201201120012100001110021202112...

66/167=0,10120000222012011200121000011100212021121011112020010212202200202101000122110110010...

67/167=0,10121111020221201002002202012122210011211221212102222000210211022101222211122010201...

68/167=0,10122221112201020110121111020221201002002202012122210011211221212102222000210211022...

69/167=0,10201101211110202212010020022020121222100112112212121022220002102110221012222111220...

70/167=0,10202212010020022020121222100112112212121022220002102110221012222111220102011012111...

71/167=0,10211022101222211122010201101211110202212010020022020121222100112112212121022220002...

72/167=0,10212202200202101000122110110010101200002220120112001210000111002120211210111120200...

73/167=0,10221012222111220102011012111102022120100200220201212221001121122121210222200021021...

74/167=0,10222200021021102210122221112201020110121111020221201002002202012122210011211221212...

75/167=0,11001010120000222012011200121000011100212021121011112020010212202200202101000122110...

76/167=0,11002120211210111120200102122022002021010001221101100101012000022201201120012100001...

77/167=0,11011001010120000222012011200121000011100212021121011112020010212202200202101000122...

78/167=0,11012111102022120100200220201212221001121122121210222200021021102210122221112201020...

79/167=0,11020221201002002202012122210011211221212102222000210211022101222211122010201101211...

80/167=0,11022101222211122010201101211110202212010020022020121222100112112212121022220002102...

81/167=0,11100212021121011112020010212202200202101000122110110010101200002220120112001210000...

82/167=0,11102022120100200220201212221001121122121210222200021021102210122221112201020110121...

83/167=0,11110202212010020022020121222100112112212121022220002102110221012222111220102011012...

84/167=0,11112020010212202200202101000122110110010101200002220120112001210000111002120211210...

85/167=0,11120200102122022002021010001221101100101012000022201201120012100001110021202112101...

86/167=0,11122010201101211110202212010020022020121222100112112212121022220002102110221012222...

87/167=0,11200121000011100212021121011112020010212202200202101000122110110010101200002220120...

88/167=0,11202001021220220020210100012211011001010120000222012011200121000011100212021121011...

89/167=0,11210111120200102122022002021010001221101100101012000022201201120012100001110021202...

90/167=0,11211221212102222000210211022101222211122010201101211110202212010020022020121222100...

91/167=0,11220102011012111102022120100200220201212221001121122121210222200021021102210122221...

92/167=0,11221212102222000210211022101222211122010201101211110202212010020022020121222100112...

93/167=0,12000022201201120012100001110021202112101111202001021220220020210100012211011001010...

94/167=0,12001210000111002120211210111120200102122022002021010001221101100101012000022201201...

95/167=0,12010020022020121222100112112212121022220002102110221012222111220102011012111102022...

96/167=0,12011200121000011100212021121011112020010212202200202101000122110110010101200002220...

97/167=0,12020010212202200202101000122110110010101200002220120112001210000111002120211210111...

98/167=0,12021121011112020010212202200202101000122110110010101200002220120112001210000111002...

99/167=0,12100001110021202112101111202001021220220020210100012211011001010120000222012011200...

100/167=0,12101111202001021220220020210100012211011001010120000222012011200121000011100212021...

101/167=0,12102222000210211022101222211122010201101211110202212010020022020121222100112112212...

102/167=0,12111102022120100200220201212221001121122121210222200021021102210122221112201020110...

103/167=0,12112212121022220002102110221012222111220102011012111102022120100200220201212221001...

104/167=0,12121022220002102110221012222111220102011012111102022120100200220201212221001121122...

105/167=0,12122210011211221212102222000210211022101222211122010201101211110202212010020022020...

106/167=0,12201020110121111020221201002002202012122210011211221212102222000210211022101222211...

107/167=0,12202200202101000122110110010101200002220120112001210000111002120211210111120200102...

108/167=0,12211011001010120000222012011200121000011100212021121011112020010212202200202101000...

109/167=0,12212121022220002102110221012222111220102011012111102022120100200220201212221001121...

110/167=0,12221001121122121210222200021021102210122221112201020110121111020221201002002202012...

111/167=0,12222111220102011012111102022120100200220201212221001121122121210222200021021102210...

112/167=0,20000222012011200121000011100212021121011112020010212202200202101000122110110010101...

113/167=0,20002102110221012222111220102011012111102022120100200220201212221001121122121210222...

114/167=0,20010212202200202101000122110110010101200002220120112001210000111002120211210111120...

115/167=0,20012100001110021202112101111202001021220220020210100012211011001010120000222012011...

116/167=0,20020210100012211011001010120000222012011200121000011100212021121011112020010212202...

117/167=0,20022020121222100112112212121022220002102110221012222111220102011012111102022120100...

118/167=0,20100200220201212221001121122121210222200021021102210122221112201020110121111020221...

119/167=0,20102011012111102022120100200220201212221001121122121210222200021021102210122221112...

120/167=0,20110121111020221201002002202012122210011211221212102222000210211022101222211122010...

121/167=0,20112001210000111002120211210111120200102122022002021010001221101100101012000022201...

122/167=0,20120112001210000111002120211210111120200102122022002021010001221101100101012000022...

123/167=0,20121222100112112212121022220002102110221012222111220102011012111102022120100200220...

124/167=0,20200102122022002021010001221101100101012000022201201120012100001110021202112101111...

125/167=0,20201212221001121122121210222200021021102210122221112201020110121111020221201002002...

126/167=0,20210100012211011001010120000222012011200121000011100212021121011112020010212202200...

127/167=0,20211210111120200102122022002021010001221101100101012000022201201120012100001110021...

128/167=0,20220020210100012211011001010120000222012011200121000011100212021121011112020010212...

129/167=0,20221201002002202012122210011211221212102222000210211022101222211122010201101211110...

130/167=0,21000011100212021121011112020010212202200202101000122110110010101200002220120112001...

131/167=0,21001121122121210222200021021102210122221112201020110121111020221201002002202012122...

132/167=0,21010001221101100101012000022201201120012100001110021202112101111202001021220220020...

133/167=0,21011112020010212202200202101000122110110010101200002220120112001210000111002120211...

134/167=0,21012222111220102011012111102022120100200220201212221001121122121210222200021021102...

135/167=0,21021102210122221112201020110121111020221201002002202012122210011211221212102222000...

136/167=0,21022220002102110221012222111220102011012111102022120100200220201212221001121122121...

137/167=0,21101100101012000022201201120012100001110021202112101111202001021220220020210100012...

138/167=0,21102210122221112201020110121111020221201002002202012122210011211221212102222000210...

139/167=0,21111020221201002002202012122210011211221212102222000210211022101222211122010201101...

140/167=0,21112201020110121111020221201002002202012122210011211221212102222000210211022101222...

141/167=0,21121011112020010212202200202101000122110110010101200002220120112001210000111002120...

142/167=0,21122121210222200021021102210122221112201020110121111020221201002002202012122210011...

143/167=0,21201002002202012122210011211221212102222000210211022101222211122010201101211110202...

144/167=0,21202112101111202001021220220020210100012211011001010120000222012011200121000011100...

145/167=0,21210222200021021102210122221112201020110121111020221201002002202012122210011211221...

146/167=0,21212102222000210211022101222211122010201101211110202212010020022020121222100112112...

147/167=0,21220220020210100012211011001010120000222012011200121000011100212021121011112020010...

148/167=0,21222100112112212121022220002102110221012222111220102011012111102022120100200220201...

149/167=0,22000210211022101222211122010201101211110202212010020022020121222100112112212121022...

150/167=0,22002021010001221101100101012000022201201120012100001110021202112101111202001021220...

151/167=0,22010201101211110202212010020022020121222100112112212121022220002102110221012222111...

152/167=0,22012011200121000011100212021121011112020010212202200202101000122110110010101200002...

153/167=0,22020121222100112112212121022220002102110221012222111220102011012111102022120100200...

154/167=0,22022002021010001221101100101012000022201201120012100001110021202112101111202001021...

155/167=0,22100112112212121022220002102110221012222111220102011012111102022120100200220201212...

156/167=0,22101222211122010201101211110202212010020022020121222100112112212121022220002102110...

157/167=0,22110110010101200002220120112001210000111002120211210111120200102122022002021010001...

158/167=0,22111220102011012111102022120100200220201212221001121122121210222200021021102210122...

159/167=0,22120100200220201212221001121122121210222200021021102210122221112201020110121111020...

160/167=0,22121210222200021021102210122221112201020110121111020221201002002202012122210011211...

161/167=0,22200021021102210122221112201020110121111020221201002002202012122210011211221212102...

162/167=0,22201201120012100001110021202112101111202001021220220020210100012211011001010120000...

163/167=0,22210011211221212102222000210211022101222211122010201101211110202212010020022020121...

164/167=0,22211122010201101211110202212010020022020121222100112112212121022220002102110221012...

165/167=0,22220002102110221012222111220102011012111102022120100200220201212221001121122121210...

166/167=0,22221112201020110121111020221201002002202012122210011211221212102222000210211022101...

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