Il existe 4 périodes de 27 chiffres pour n/109 en base 3.

Pour toutes les fractions de n/109 en base 3, la période de 1/109 revient alors 27 fois (en orange)

1/109=0,000020200120121100102010222...

2/109=0,000111101011012200211021221...

3/109=0,000202001201211001020102220...

4/109=0,000222202022102101122120212...

5/109=0,001020102220000202001201211...

6/109=0,001111010110122002110212210...

7/109=0,001201211001020102220000202...

8/109=0,001222111121211210022011201...

9/109=0,002020012012110010201022200...

10/109=0,002110212210001111010110122...

11/109=0,002201120100122211112121121...

12/109=0,002222020221021011221202120...

13/109=0,010012221111212112100220112...

14/109=0,010110122002110212210001111...

15/109=0,010201022200002020012012110...

16/109=0,010222000020200120121100102...

17/109=0,011012200211021221000111101...

18/109=0,011110101101220021102122100...

19/109=0,011201001222111121211210022...

20/109=0,011221202120002222020221021...

21/109=0,012012110010201022200002020...

22/109=0,012110010201022200002020012...

23/109=0,012200211021221000111101011...

24/109=0,012221111212112100220112010...

25/109=0,020012012110010201022200002...

26/109=0,020102220000202001201211001...

27/109=0,020200120121100102010222000...

28/109=0,020221021011221202120002222...

29/109=0,021011221202120002222020221...

30/109=0,021102122100011110101101220...

31/109=0,021200022220202210210112212...

32/109=0,021221000111101011012200211...

33/109=0,022011201001222111121211210...

34/109=0,022102101122120212000222202...

35/109=0,022200002020012012110010201...

36/109=0,022220202210210112212021200...

37/109=0,100011110101101220021102122...

38/109=0,100102010222000020200120121...

39/109=0,100122211112121121002201120...

40/109=0,100220112010012221111212112...

41/109=0,101011012200211021221000111...

42/109=0,101101220021102122100011110...

43/109=0,101122120212000222202022102...

44/109=0,101220021102122100011110101...

45/109=0,102010222000020200120121100...

46/109=0,102101122120212000222202022...

47/109=0,102122100011110101101220021...

48/109=0,102220000202001201211001020...

49/109=0,110010201022200002020012012...

50/109=0,110101101220021102122100011...

51/109=0,110122002110212210001111010...

52/109=0,110212210001111010110122002...

53/109=0,111010110122002110212210001...

54/109=0,111101011012200211021221000...

55/109=0,111121211210022011201001222...

56/109=0,111212112100220112010012221...

57/109=0,112010012221111212112100220...

58/109=0,112100220112010012221111212...

59/109=0,112121121002201120100122211...

60/109=0,112212021200022220202210210...

61/109=0,120002222020221021011221202...

62/109=0,120100122211112121121002201...

63/109=0,120121100102010222000020200...

64/109=0,120212000222202022102101122...

65/109=0,121002201120100122211112121...

66/109=0,121100102010222000020200120...

67/109=0,121121002201120100122211112...

68/109=0,121211210022011201001222111...

69/109=0,122002110212210001111010110...

70/109=0,122100011110101101220021102...

71/109=0,122120212000222202022102101...

72/109=0,122211112121121002201120100...

73/109=0,200002020012012110010201022...

74/109=0,200022220202210210112212021...

75/109=0,200120121100102010222000020...

76/109=0,200211021221000111101011012...

77/109=0,201001222111121211210022011...

78/109=0,201022200002020012012110010...

79/109=0,201120100122211112121121002...

80/109=0,201211001020102220000202001...

81/109=0,202001201211001020102220000...

82/109=0,202022102101122120212000222...

83/109=0,202120002222020221021011221...

84/109=0,202210210112212021200022220...

85/109=0,210001111010110122002110212...

86/109=0,210022011201001222111121211...

87/109=0,210112212021200022220202210...

88/109=0,210210112212021200022220202...

89/109=0,211001020102220000202001201...

90/109=0,211021221000111101011012200...

91/109=0,211112121121002201120100122...

92/109=0,211210022011201001222111121...

93/109=0,212000222202022102101122120...

94/109=0,212021200022220202210210112...

95/109=0,212112100220112010012221111...

96/109=0,212210001111010110122002110...

97/109=0,220000202001201211001020102...

98/109=0,220021102122100011110101101...

99/109=0,220112010012221111212112100...

100/109=0,220202210210112212021200022...

101/109=0,221000111101011012200211021...

102/109=0,221021011221202120002222020...

103/109=0,221111212112100220112010012...

104/109=0,221202120002222020221021011...

105/109=0,222000020200120121100102010...

106/109=0,222020221021011221202120002...

107/109=0,222111121211210022011201001...

108/109=0,222202022102101122120212000...

On remarque que le produit du nombre de périodes (4) et de leurs longueurs (27) est égal à 108 et donc au premier -1.