Il existe 2 périodes de 44 chiffres pour n/89 en base 5.

Pour toutes les fractions de n/89 en base 5, la période de 1/89 revient alors 44 fois (en orange)

1/89=0,00120024010302110422134432442043414233402231...

2/89=0,00240103021104221344324420434142334022310012...

3/89=0,00410132031411332322014403431241303311212243...

4/89=0,01030211042213443244204341423340223100120024...

5/89=0,01200240103021104221344324420434142334022310...

6/89=0,01320314113323220144034312413033112122430041...

7/89=0,01440343124130331121224300410132031411332322...

8/89=0,02110422134432442043414233402231001200240103...

9/89=0,02231001200240103021104221344324420434142334...

10/89=0,02401030211042213443244204341423340223100120...

11/89=0,03021104221344324420434142334022310012002401...

12/89=0,03141133232201440343124130331121224300410132...

13/89=0,03311212243004101320314113323220144034312413...

14/89=0,03431241303311212243004101320314113323220144...

15/89=0,04101320314113323220144034312413033112122430...

16/89=0,04221344324420434142334022310012002401030211...

17/89=0,04341423340223100120024010302110422134432442...

18/89=0,10012002401030211042213443244204341423340223...

19/89=0,10132031411332322014403431241303311212243004...

20/89=0,10302110422134432442043414233402231001200240...

21/89=0,10422134432442043414233402231001200240103021...

22/89=0,11042213443244204341423340223100120024010302...

23/89=0,11212243004101320314113323220144034312413033...

24/89=0,11332322014403431241303311212243004101320314...

25/89=0,12002401030211042213443244204341423340223100...

26/89=0,12122430041013203141133232201440343124130331...

27/89=0,12243004101320314113323220144034312413033112...

28/89=0,12413033112122430041013203141133232201440343...

29/89=0,13033112122430041013203141133232201440343124...

30/89=0,13203141133232201440343124130331121224300410...

31/89=0,13323220144034312413033112122430041013203141...

32/89=0,13443244204341423340223100120024010302110422...

33/89=0,14113323220144034312413033112122430041013203...

34/89=0,14233402231001200240103021104221344324420434...

35/89=0,14403431241303311212243004101320314113323220...

36/89=0,20024010302110422134432442043414233402231001...

37/89=0,20144034312413033112122430041013203141133232...

38/89=0,20314113323220144034312413033112122430041013...

39/89=0,20434142334022310012002401030211042213443244...

40/89=0,21104221344324420434142334022310012002401030...

41/89=0,21224300410132031411332322014403431241303311...

42/89=0,21344324420434142334022310012002401030211042...

43/89=0,22014403431241303311212243004101320314113323...

44/89=0,22134432442043414233402231001200240103021104...

45/89=0,22310012002401030211042213443244204341423340...

46/89=0,22430041013203141133232201440343124130331121...

47/89=0,23100120024010302110422134432442043414233402...

48/89=0,23220144034312413033112122430041013203141133...

49/89=0,23340223100120024010302110422134432442043414...

50/89=0,24010302110422134432442043414233402231001200...

51/89=0,24130331121224300410132031411332322014403431...

52/89=0,24300410132031411332322014403431241303311212...

53/89=0,24420434142334022310012002401030211042213443...

54/89=0,30041013203141133232201440343124130331121224...

55/89=0,30211042213443244204341423340223100120024010...

56/89=0,30331121224300410132031411332322014403431241...

57/89=0,31001200240103021104221344324420434142334022...

58/89=0,31121224300410132031411332322014403431241303...

59/89=0,31241303311212243004101320314113323220144034...

60/89=0,31411332322014403431241303311212243004101320...

61/89=0,32031411332322014403431241303311212243004101...

62/89=0,32201440343124130331121224300410132031411332...

63/89=0,32322014403431241303311212243004101320314113...

64/89=0,32442043414233402231001200240103021104221344...

65/89=0,33112122430041013203141133232201440343124130...

66/89=0,33232201440343124130331121224300410132031411...

67/89=0,33402231001200240103021104221344324420434142...

68/89=0,34022310012002401030211042213443244204341423...

69/89=0,34142334022310012002401030211042213443244204...

70/89=0,34312413033112122430041013203141133232201440...

71/89=0,34432442043414233402231001200240103021104221...

72/89=0,40103021104221344324420434142334022310012002...

73/89=0,40223100120024010302110422134432442043414233...

74/89=0,40343124130331121224300410132031411332322014...

75/89=0,41013203141133232201440343124130331121224300...

76/89=0,41133232201440343124130331121224300410132031...

77/89=0,41303311212243004101320314113323220144034312...

78/89=0,41423340223100120024010302110422134432442043...

79/89=0,42043414233402231001200240103021104221344324...

80/89=0,42213443244204341423340223100120024010302110...

81/89=0,42334022310012002401030211042213443244204341...

82/89=0,43004101320314113323220144034312413033112122...

83/89=0,43124130331121224300410132031411332322014403...

84/89=0,43244204341423340223100120024010302110422134...

85/89=0,43414233402231001200240103021104221344324420...

86/89=0,44034312413033112122430041013203141133232201...

87/89=0,44204341423340223100120024010302110422134432...

88/89=0,44324420434142334022310012002401030211042213...

On remarque que le produit du nombre de périodes (2) et de leurs longueurs (44) est égal à 88 et donc au premier -1.