Il existe 3 périodes de 36 chiffres pour n/109 en base 2.

Pour toutes les fractions de n/109 en base 2, la période de 1/109 revient alors 36 fois (en orange)

1/109=0,000000100101100100111111011010011011...

2/109=0,000001001011001001111110110100110110...

3/109=0,000001110000101110111110001111010001...

4/109=0,000010010110010011111101101001101100...

5/109=0,000010111011111000111101000100000111...

6/109=0,000011100001011101111100011110100010...

7/109=0,000100000111000010111011111000111101...

8/109=0,000100101100100111111011010011011000...

9/109=0,000101010010001100111010101101110011...

10/109=0,000101110111110001111010001000001110...

11/109=0,000110011101010110111001100010101001...

12/109=0,000111000010111011111000111101000100...

13/109=0,000111101000100000111000010111011111...

14/109=0,001000001110000101110111110001111010...

15/109=0,001000110011101010110111001100010101...

16/109=0,001001011001001111110110100110110000...

17/109=0,001001111110110100110110000001001011...

18/109=0,001010100100011001110101011011100110...

19/109=0,001011001001111110110100110110000001...

20/109=0,001011101111100011110100010000011100...

21/109=0,001100010101001000110011101010110111...

22/109=0,001100111010101101110011000101010010...

23/109=0,001101100000010010110010011111101101...

24/109=0,001110000101110111110001111010001000...

25/109=0,001110101011011100110001010100100011...

26/109=0,001111010001000001110000101110111110...

27/109=0,001111110110100110110000001001011001...

28/109=0,010000011100001011101111100011110100...

29/109=0,010001000001110000101110111110001111...

30/109=0,010001100111010101101110011000101010...

31/109=0,010010001100111010101101110011000101...

32/109=0,010010110010011111101101001101100000...

33/109=0,010011011000000100101100100111111011...

34/109=0,010011111101101001101100000010010110...

35/109=0,010100100011001110101011011100110001...

36/109=0,010101001000110011101010110111001100...

37/109=0,010101101110011000101010010001100111...

38/109=0,010110010011111101101001101100000010...

39/109=0,010110111001100010101001000110011101...

40/109=0,010111011111000111101000100000111000...

41/109=0,011000000100101100100111111011010011...

42/109=0,011000101010010001100111010101101110...

43/109=0,011001001111110110100110110000001001...

44/109=0,011001110101011011100110001010100100...

45/109=0,011010011011000000100101100100111111...

46/109=0,011011000000100101100100111111011010...

47/109=0,011011100110001010100100011001110101...

48/109=0,011100001011101111100011110100010000...

49/109=0,011100110001010100100011001110101011...

50/109=0,011101010110111001100010101001000110...

51/109=0,011101111100011110100010000011100001...

52/109=0,011110100010000011100001011101111100...

53/109=0,011111000111101000100000111000010111...

54/109=0,011111101101001101100000010010110010...

55/109=0,100000010010110010011111101101001101...

56/109=0,100000111000010111011111000111101000...

57/109=0,100001011101111100011110100010000011...

58/109=0,100010000011100001011101111100011110...

59/109=0,100010101001000110011101010110111001...

60/109=0,100011001110101011011100110001010100...

61/109=0,100011110100010000011100001011101111...

62/109=0,100100011001110101011011100110001010...

63/109=0,100100111111011010011011000000100101...

64/109=0,100101100100111111011010011011000000...

65/109=0,100110001010100100011001110101011011...

66/109=0,100110110000001001011001001111110110...

67/109=0,100111010101101110011000101010010001...

68/109=0,100111111011010011011000000100101100...

69/109=0,101000100000111000010111011111000111...

70/109=0,101001000110011101010110111001100010...

71/109=0,101001101100000010010110010011111101...

72/109=0,101010010001100111010101101110011000...

73/109=0,101010110111001100010101001000110011...

74/109=0,101011011100110001010100100011001110...

75/109=0,101100000010010110010011111101101001...

76/109=0,101100100111111011010011011000000100...

77/109=0,101101001101100000010010110010011111...

78/109=0,101101110011000101010010001100111010...

79/109=0,101110011000101010010001100111010101...

80/109=0,101110111110001111010001000001110000...

81/109=0,101111100011110100010000011100001011...

82/109=0,110000001001011001001111110110100110...

83/109=0,110000101110111110001111010001000001...

84/109=0,110001010100100011001110101011011100...

85/109=0,110001111010001000001110000101110111...

86/109=0,110010011111101101001101100000010010...

87/109=0,110011000101010010001100111010101101...

88/109=0,110011101010110111001100010101001000...

89/109=0,110100010000011100001011101111100011...

90/109=0,110100110110000001001011001001111110...

91/109=0,110101011011100110001010100100011001...

92/109=0,110110000001001011001001111110110100...

93/109=0,110110100110110000001001011001001111...

94/109=0,110111001100010101001000110011101010...

95/109=0,110111110001111010001000001110000101...

96/109=0,111000010111011111000111101000100000...

97/109=0,111000111101000100000111000010111011...

98/109=0,111001100010101001000110011101010110...

99/109=0,111010001000001110000101110111110001...

100/109=0,111010101101110011000101010010001100...

101/109=0,111011010011011000000100101100100111...

102/109=0,111011111000111101000100000111000010...

103/109=0,111100011110100010000011100001011101...

104/109=0,111101000100000111000010111011111000...

105/109=0,111101101001101100000010010110010011...

106/109=0,111110001111010001000001110000101110...

107/109=0,111110110100110110000001001011001001...

108/109=0,111111011010011011000000100101100100...

On remarque que le produit du nombre de périodes (3) et de leurs longueurs (36) est égal à 108 et donc au premier -1.