Il existe 2 périodes de 78 chiffres pour n/157 en base 3.

Pour toutes les fractions de n/157 en base 3, la période de 1/157 revient alors 78 fois (en orange)

1/157=0,000011122100222022102112002202021220120222211100122000200120110220020201002102...

2/157=0,000100021201221121212001012111120211011222122201021001101010221210111102011211...

3/157=0,000111221002220221021120022020212201202222111001220002001201102200202010021020...

4/157=0,000200120110220020201002102000011122100222022102112002202021220120222211100122...

5/157=0,000212012211212120010121111202110112221222010210011010102212101111020112110001...

6/157=0,001000212012211212120010121111202110112221222010210011010102212101111020112110...

7/157=0,001012111120211011222122201021001101010221210111102011211000100021201221121212...

8/157=0,001101010221210111102011211000100021201221121212001012111120211011222122201021...

9/157=0,001112210022202210211200220202122012022221110012200020012011022002020100210200...

10/157=0,001201102200202010021020000111221002220221021120022020212201202222111001220002...

11/157=0,001220002001201102200202010021020000111221002220221021120022020212201202222111...

12/157=0,002001201102200202010021020000111221002220221021120022020212201202222111001220...

13/157=0,002020100210200001112210022202210211200220202122012022221110012200020012011022...

14/157=0,002102000011122100222022102112002202021220120222211100122000200120110220020201...

15/157=0,002120122112121200101211112021101122212220102100110101022121011110201121100010...

16/157=0,002202021220120222211100122000200120110220020201002102000011122100222022102112...

17/157=0,002220221021120022020212201202222111001220002001201102200202010021020000111221...

18/157=0,010002120122112121200101211112021101122212220102100110101022121011110201121100...

19/157=0,010021020000111221002220221021120022020212201202222111001220002001201102200202...

20/157=0,010102212101111020112110001000212012211212120010121111202110112221222010210011...

21/157=0,010121111202110112221222010210011010102212101111020112110001000212012211212120...

22/157=0,010210011010102212101111020112110001000212012211212120010121111202110112221222...

23/157=0,010221210111102011211000100021201221121212001012111120211011222122201021001101...

24/157=0,011010102212101111020112110001000212012211212120010121111202110112221222010210...

25/157=0,011022002020100210200001112210022202210211200220202122012022221110012200020012...

26/157=0,011110201121100010002120122112121200101211112021101122212220102100110101022121...

27/157=0,011122100222022102112002202021220120222211100122000200120110220020201002102000...

28/157=0,011211000100021201221121212001012111120211011222122201021001101010221210111102...

29/157=0,011222122201021001101010221210111102011211000100021201221121212001012111120211...

30/157=0,012011022002020100210200001112210022202210211200220202122012022221110012200020...

31/157=0,012022221110012200020012011022002020100210200001112210022202210211200220202122...

32/157=0,012111120211011222122201021001101010221210111102011211000100021201221121212001...

33/157=0,012200020012011022002020100210200001112210022202210211200220202122012022221110...

34/157=0,012211212120010121111202110112221222010210011010102212101111020112110001000212...

35/157=0,020000111221002220221021120022020212201202222111001220002001201102200202010021...

36/157=0,020012011022002020100210200001112210022202210211200220202122012022221110012200...

37/157=0,020100210200001112210022202210211200220202122012022221110012200020012011022002...

38/157=0,020112110001000212012211212120010121111202110112221222010210011010102212101111...

39/157=0,020201002102000011122100222022102112002202021220120222211100122000200120110220...

40/157=0,020212201202222111001220002001201102200202010021020000111221002220221021120022...

41/157=0,021001101010221210111102011211000100021201221121212001012111120211011222122201...

42/157=0,021020000111221002220221021120022020212201202222111001220002001201102200202010...

43/157=0,021101122212220102100110101022121011110201121100010002120122112121200101211112...

44/157=0,021120022020212201202222111001220002001201102200202010021020000111221002220221...

45/157=0,021201221121212001012111120211011222122201021001101010221210111102011211000100...

46/157=0,021220120222211100122000200120110220020201002102000011122100222022102112002202...

47/157=0,022002020100210200001112210022202210211200220202122012022221110012200020012011...

48/157=0,022020212201202222111001220002001201102200202010021020000111221002220221021120...

49/157=0,022102112002202021220120222211100122000200120110220020201002102000011122100222...

50/157=0,022121011110201121100010002120122112121200101211112021101122212220102100110101...

51/157=0,022202210211200220202122012022221110012200020012011022002020100210200001112210...

52/157=0,022221110012200020012011022002020100210200001112210022202210211200220202122012...

53/157=0,100010002120122112121200101211112021101122212220102100110101022121011110201121...

54/157=0,100021201221121212001012111120211011222122201021001101010221210111102011211000...

55/157=0,100110101022121011110201121100010002120122112121200101211112021101122212220102...

56/157=0,100122000200120110220020201002102000011122100222022102112002202021220120222211...

57/157=0,100210200001112210022202210211200220202122012022221110012200020012011022002020...

58/157=0,100222022102112002202021220120222211100122000200120110220020201002102000011122...

59/157=0,101010221210111102011211000100021201221121212001012111120211011222122201021001...

60/157=0,101022121011110201121100010002120122112121200101211112021101122212220102100110...

61/157=0,101111020112110001000212012211212120010121111202110112221222010210011010102212...

62/157=0,101122212220102100110101022121011110201121100010002120122112121200101211112021...

63/157=0,101211112021101122212220102100110101022121011110201121100010002120122112121200...

64/157=0,102000011122100222022102112002202021220120222211100122000200120110220020201002...

65/157=0,102011211000100021201221121212001012111120211011222122201021001101010221210111...

66/157=0,102100110101022121011110201121100010002120122112121200101211112021101122212220...

67/157=0,102112002202021220120222211100122000200120110220020201002102000011122100222022...

68/157=0,102200202010021020000111221002220221021120022020212201202222111001220002001201...

69/157=0,102212101111020112110001000212012211212120010121111202110112221222010210011010...

70/157=0,110001000212012211212120010121111202110112221222010210011010102212101111020112...

71/157=0,110012200020012011022002020100210200001112210022202210211200220202122012022221...

72/157=0,110101022121011110201121100010002120122112121200101211112021101122212220102100...

73/157=0,110112221222010210011010102212101111020112110001000212012211212120010121111202...

74/157=0,110201121100010002120122112121200101211112021101122212220102100110101022121011...

75/157=0,110220020201002102000011122100222022102112002202021220120222211100122000200120...

76/157=0,111001220002001201102200202010021020000111221002220221021120022020212201202222...

77/157=0,111020112110001000212012211212120010121111202110112221222010210011010102212101...

78/157=0,111102011211000100021201221121212001012111120211011222122201021001101010221210...

79/157=0,111120211011222122201021001101010221210111102011211000100021201221121212001012...

80/157=0,111202110112221222010210011010102212101111020112110001000212012211212120010121...

81/157=0,111221002220221021120022020212201202222111001220002001201102200202010021020000...

82/157=0,112002202021220120222211100122000200120110220020201002102000011122100222022102...

83/157=0,112021101122212220102100110101022121011110201121100010002120122112121200101211...

84/157=0,112110001000212012211212120010121111202110112221222010210011010102212101111020...

85/157=0,112121200101211112021101122212220102100110101022121011110201121100010002120122...

86/157=0,112210022202210211200220202122012022221110012200020012011022002020100210200001...

87/157=0,112221222010210011010102212101111020112110001000212012211212120010121111202110...

88/157=0,120010121111202110112221222010210011010102212101111020112110001000212012211212...

89/157=0,120022020212201202222111001220002001201102200202010021020000111221002220221021...

90/157=0,120110220020201002102000011122100222022102112002202021220120222211100122000200...

91/157=0,120122112121200101211112021101122212220102100110101022121011110201121100010002...

92/157=0,120211011222122201021001101010221210111102011211000100021201221121212001012111...

93/157=0,120222211100122000200120110220020201002102000011122100222022102112002202021220...

94/157=0,121011110201121100010002120122112121200101211112021101122212220102100110101022...

95/157=0,121100010002120122112121200101211112021101122212220102100110101022121011110201...

96/157=0,121111202110112221222010210011010102212101111020112110001000212012211212120010...

97/157=0,121200101211112021101122212220102100110101022121011110201121100010002120122112...

98/157=0,121212001012111120211011222122201021001101010221210111102011211000100021201221...

99/157=0,122000200120110220020201002102000011122100222022102112002202021220120222211100...

100/157=0,122012022221110012200020012011022002020100210200001112210022202210211200220202...

101/157=0,122100222022102112002202021220120222211100122000200120110220020201002102000011...

102/157=0,122112121200101211112021101122212220102100110101022121011110201121100010002120...

103/157=0,122201021001101010221210111102011211000100021201221121212001012111120211011222...

104/157=0,122212220102100110101022121011110201121100010002120122112121200101211112021101...

105/157=0,200001112210022202210211200220202122012022221110012200020012011022002020100210...

106/157=0,200020012011022002020100210200001112210022202210211200220202122012022221110012...

107/157=0,200101211112021101122212220102100110101022121011110201121100010002120122112121...

108/157=0,200120110220020201002102000011122100222022102112002202021220120222211100122000...

109/157=0,200202010021020000111221002220221021120022020212201202222111001220002001201102...

110/157=0,200220202122012022221110012200020012011022002020100210200001112210022202210211...

111/157=0,201002102000011122100222022102112002202021220120222211100122000200120110220020...

112/157=0,201021001101010221210111102011211000100021201221121212001012111120211011222122...

113/157=0,201102200202010021020000111221002220221021120022020212201202222111001220002001...

114/157=0,201121100010002120122112121200101211112021101122212220102100110101022121011110...

115/157=0,201202222111001220002001201102200202010021020000111221002220221021120022020212...

116/157=0,201221121212001012111120211011222122201021001101010221210111102011211000100021...

117/157=0,202010021020000111221002220221021120022020212201202222111001220002001201102200...

118/157=0,202021220120222211100122000200120110220020201002102000011122100222022102112002...

119/157=0,202110112221222010210011010102212101111020112110001000212012211212120010121111...

120/157=0,202122012022221110012200020012011022002020100210200001112210022202210211200220...

121/157=0,202210211200220202122012022221110012200020012011022002020100210200001112210022...

122/157=0,202222111001220002001201102200202010021020000111221002220221021120022020212201...

123/157=0,210011010102212101111020112110001000212012211212120010121111202110112221222010...

124/157=0,210022202210211200220202122012022221110012200020012011022002020100210200001112...

125/157=0,210111102011211000100021201221121212001012111120211011222122201021001101010221...

126/157=0,210200001112210022202210211200220202122012022221110012200020012011022002020100...

127/157=0,210211200220202122012022221110012200020012011022002020100210200001112210022202...

128/157=0,211000100021201221121212001012111120211011222122201021001101010221210111102011...

129/157=0,211011222122201021001101010221210111102011211000100021201221121212001012111120...

130/157=0,211100122000200120110220020201002102000011122100222022102112002202021220120222...

131/157=0,211112021101122212220102100110101022121011110201121100010002120122112121200101...

132/157=0,211200220202122012022221110012200020012011022002020100210200001112210022202210...

133/157=0,211212120010121111202110112221222010210011010102212101111020112110001000212012...

134/157=0,212001012111120211011222122201021001101010221210111102011211000100021201221121...

135/157=0,212012211212120010121111202110112221222010210011010102212101111020112110001000...

136/157=0,212101111020112110001000212012211212120010121111202110112221222010210011010102...

137/157=0,212120010121111202110112221222010210011010102212101111020112110001000212012211...

138/157=0,212201202222111001220002001201102200202010021020000111221002220221021120022020...

139/157=0,212220102100110101022121011110201121100010002120122112121200101211112021101122...

140/157=0,220002001201102200202010021020000111221002220221021120022020212201202222111001...

141/157=0,220020201002102000011122100222022102112002202021220120222211100122000200120110...

142/157=0,220102100110101022121011110201121100010002120122112121200101211112021101122212...

143/157=0,220120222211100122000200120110220020201002102000011122100222022102112002202021...

144/157=0,220202122012022221110012200020012011022002020100210200001112210022202210211200...

145/157=0,220221021120022020212201202222111001220002001201102200202010021020000111221002...

146/157=0,221002220221021120022020212201202222111001220002001201102200202010021020000111...

147/157=0,221021120022020212201202222111001220002001201102200202010021020000111221002220...

148/157=0,221110012200020012011022002020100210200001112210022202210211200220202122012022...

149/157=0,221121212001012111120211011222122201021001101010221210111102011211000100021201...

150/157=0,221210111102011211000100021201221121212001012111120211011222122201021001101010...

151/157=0,221222010210011010102212101111020112110001000212012211212120010121111202110112...

152/157=0,222010210011010102212101111020112110001000212012211212120010121111202110112221...

153/157=0,222022102112002202021220120222211100122000200120110220020201002102000011122100...

154/157=0,222111001220002001201102200202010021020000111221002220221021120022020212201202...

155/157=0,222122201021001101010221210111102011211000100021201221121212001012111120211011...

156/157=0,222211100122000200120110220020201002102000011122100222022102112002202021220120...

On remarque que le produit du nombre de périodes (2) et de leurs longueurs (78) est égal à 156 et donc au premier -1.