Il existe 2 périodes de 41 chiffres pour n/83 en base 3.

Pour toutes les fractions de n/83 en base 3, la période de 1/83 revient alors 41 fois (en orange)

1/83=0,00002221001022010120121120221101002021111...

2/83=0,00012212002121021011020011212202011112222...

3/83=0,00022210010220101201211202211010020211110...

4/83=0,00102201012012112022110100202111100002221...

5/83=0,00112122020111122220001221200212102101102...

6/83=0,00122120021210210110200112122020111122220...

7/83=0,00202111100002221001022010120121120221101...

8/83=0,00212102101102001121220201111222200012212...

9/83=0,00222100102201012012112022110100202111100...

10/83=0,01002021111000022210010220101201211202211...

11/83=0,01012012112022110100202111100002221001022...

12/83=0,01022010120121120221101002021111000022210...

13/83=0,01102001121220201111222200012212002121021...

14/83=0,01111222200012212002121021011020011212202...

15/83=0,01121220201111222200012212002121021011020...

16/83=0,01201211202211010020211110000222100102201...

17/83=0,01211202211010020211110000222100102201012...

18/83=0,01221200212102101102001121220201111222200...

19/83=0,02001121220201111222200012212002121021011...

20/83=0,02011112222000122120021210210110200112122...

21/83=0,02021111000022210010220101201211202211010...

22/83=0,02101102001121220201111222200012212002121...

23/83=0,02111100002221001022010120121120221101002...

24/83=0,02121021011020011212202011112222000122120...

25/83=0,02201012012112022110100202111100002221001...

26/83=0,02211010020211110000222100102201012012112...

27/83=0,02221001022010120121120221101002021111000...

28/83=0,10000222100102201012012112022110100202111...

29/83=0,10010220101201211202211010020211110000222...

30/83=0,10020211110000222100102201012012112022110...

31/83=0,10100202111100002221001022010120121120221...

32/83=0,10110200112122020111122220001221200212102...

33/83=0,10120121120221101002021111000022210010220...

34/83=0,10200112122020111122220001221200212102101...

35/83=0,10210110200112122020111122220001221200212...

36/83=0,10220101201211202211010020211110000222100...

37/83=0,11000022210010220101201211202211010020211...

38/83=0,11010020211110000222100102201012012112022...

39/83=0,11020011212202011112222000122120021210210...

40/83=0,11100002221001022010120121120221101002021...

41/83=0,11110000222100102201012012112022110100202...

42/83=0,11112222000122120021210210110200112122020...

43/83=0,11122220001221200212102101102001121220201...

44/83=0,11202211010020211110000222100102201012012...

45/83=0,11212202011112222000122120021210210110200...

46/83=0,11222200012212002121021011020011212202011...

47/83=0,12002121021011020011212202011112222000122...

48/83=0,12012112022110100202111100002221001022010...

49/83=0,12022110100202111100002221001022010120121...

50/83=0,12102101102001121220201111222200012212002...

51/83=0,12112022110100202111100002221001022010120...

52/83=0,12122020111122220001221200212102101102001...

53/83=0,12202011112222000122120021210210110200112...

54/83=0,12212002121021011020011212202011112222000...

55/83=0,12222000122120021210210110200112122020111...

56/83=0,20001221200212102101102001121220201111222...

57/83=0,20011212202011112222000122120021210210110...

58/83=0,20021210210110200112122020111122220001221...

59/83=0,20101201211202211010020211110000222100102...

60/83=0,20111122220001221200212102101102001121220...

61/83=0,20121120221101002021111000022210010220101...

62/83=0,20201111222200012212002121021011020011212...

63/83=0,20211110000222100102201012012112022110100...

64/83=0,20221101002021111000022210010220101201211...

65/83=0,21001022010120121120221101002021111000022...

66/83=0,21011020011212202011112222000122120021210...

67/83=0,21021011020011212202011112222000122120021...

68/83=0,21101002021111000022210010220101201211202...

69/83=0,21111000022210010220101201211202211010020...

70/83=0,21120221101002021111000022210010220101201...

71/83=0,21200212102101102001121220201111222200012...

72/83=0,21210210110200112122020111122220001221200...

73/83=0,21220201111222200012212002121021011020011...

74/83=0,22000122120021210210110200112122020111122...

75/83=0,22010120121120221101002021111000022210010...

76/83=0,22020111122220001221200212102101102001121...

77/83=0,22100102201012012112022110100202111100002...

78/83=0,22110100202111100002221001022010120121120...

79/83=0,22120021210210110200112122020111122220001...

80/83=0,22200012212002121021011020011212202011112...

81/83=0,22210010220101201211202211010020211110000...

82/83=0,22220001221200212102101102001121220201111...

On remarque que le produit du nombre de périodes (2) et de leurs longueurs (41) est égal à 82 et donc au premier -1.