Il existe 8 périodes de 11 chiffres pour n/89 en base 4.

Pour toutes les fractions de n/89 en base 4, la période de 1/89 revient alors 11 fois (en orange)

1/89=0,00023200113...

2/89=0,00113000232...

3/89=0,00202201011...

4/89=0,00232001130...

5/89=0,00321201303...

6/89=0,01011002022...

7/89=0,01100202201...

8/89=0,01130002320...

9/89=0,01213203033...

10/89=0,01303003212...

11/89=0,01332203331...

12/89=0,02022010110...

13/89=0,02111210223...

14/89=0,02201011002...

15/89=0,02230211121...

16/89=0,02320011300...

17/89=0,03003212013...

18/89=0,03033012132...

19/89=0,03122212311...

20/89=0,03212013030...

21/89=0,03301213203...

22/89=0,03331013322...

23/89=0,10020220101...

24/89=0,10110020220...

25/89=0,10133220333...

26/89=0,10223021112...

27/89=0,10312221231...

28/89=0,11002022010...

29/89=0,11031222123...

30/89=0,11121022302...

31/89=0,11210223021...

32/89=0,11300023200...

33/89=0,11323223313...

34/89=0,12013030032...

35/89=0,12102230211...

36/89=0,12132030330...

37/89=0,12221231103...

38/89=0,12311031222...

39/89=0,13000232001...

40/89=0,13030032120...

41/89=0,13113232233...

42/89=0,13203033012...

43/89=0,13232233131...

44/89=0,13322033310...

45/89=0,20011300023...

46/89=0,20101100202...

47/89=0,20130300321...

48/89=0,20220101100...

49/89=0,20303301213...

50/89=0,20333101332...

51/89=0,21022302111...

52/89=0,21112102230...

53/89=0,21201303003...

54/89=0,21231103122...

55/89=0,21320303301...

56/89=0,22010110020...

57/89=0,22033310133...

58/89=0,22123110312...

59/89=0,22212311031...

60/89=0,22302111210...

61/89=0,22331311323...

62/89=0,23021112102...

63/89=0,23110312221...

64/89=0,23200113000...

65/89=0,23223313113...

66/89=0,23313113232...

67/89=0,30002320011...

68/89=0,30032120130...

69/89=0,30121320303...

70/89=0,30211121022...

71/89=0,30300321201...

72/89=0,30330121320...

73/89=0,31013322033...

74/89=0,31103122212...

75/89=0,31132322331...

76/89=0,31222123110...

77/89=0,31311323223...

78/89=0,32001130002...

79/89=0,32030330121...

80/89=0,32120130300...

81/89=0,32203331013...

82/89=0,32233131132...

83/89=0,32322331311...

84/89=0,33012132030...

85/89=0,33101332203...

86/89=0,33131132322...

87/89=0,33220333101...

88/89=0,33310133220...

On remarque que le produit du nombre de périodes (8) et de leurs longueurs (11) est égal à 88 et donc au premier -1.