Il existe 2 périodes de 29 chiffres pour n/59 en base 3.

Pour toutes les fractions de n/59 en base 3, la période de 1/59 revient alors 29 fois (en orange)

1/59=0,00011010012111021022200202121...

2/59=0,00022020101222112122101112012...

3/59=0,00110100121110210222002021210...

4/59=0,00121110210222002021210001101...

5/59=0,00202121000110100121110210222...

6/59=0,00220201012221121221011120120...

7/59=0,01001211102102220020212100011...

8/59=0,01012221121221011120120002202...

9/59=0,01101001211102102220020212100...

10/59=0,01112012000220201012221121221...

11/59=0,01200022020101222112122101112...

12/59=0,01211102102220020212100011010...

13/59=0,01222112122101112012000220201...

14/59=0,02010122211212210111201200022...

15/59=0,02021210001101001211102102220...

16/59=0,02102220020212100011010012111...

17/59=0,02121000110100121110210222002...

18/59=0,02202010122211212210111201200...

19/59=0,02220020212100011010012111021...

20/59=0,10001101001211102102220020212...

21/59=0,10012111021022200202121000110...

22/59=0,10100121110210222002021210001...

23/59=0,10111201200022020101222112122...

24/59=0,10122211212210111201200022020...

25/59=0,10210222002021210001101001211...

26/59=0,10222002021210001101001211102...

27/59=0,11010012111021022200202121000...

28/59=0,11021022200202121000110100121...

29/59=0,11102102220020212100011010012...

30/59=0,11120120002202010122211212210...

31/59=0,11201200022020101222112122101...

32/59=0,11212210111201200022020101222...

33/59=0,12000220201012221121221011120...

34/59=0,12012000220201012221121221011...

35/59=0,12100011010012111021022200202...

36/59=0,12111021022200202121000110100...

37/59=0,12122101112012000220201012221...

38/59=0,12210111201200022020101222112...

39/59=0,12221121221011120120002202010...

40/59=0,20002202010122211212210111201...

41/59=0,20020212100011010012111021022...

42/59=0,20101222112122101112012000220...

43/59=0,20120002202010122211212210111...

44/59=0,20201012221121221011120120002...

45/59=0,20212100011010012111021022200...

46/59=0,21000110100121110210222002021...

47/59=0,21011120120002202010122211212...

48/59=0,21022200202121000110100121110...

49/59=0,21110210222002021210001101001...

50/59=0,21121221011120120002202010122...

51/59=0,21210001101001211102102220020...

52/59=0,21221011120120002202010122211...

53/59=0,22002021210001101001211102102...

54/59=0,22020101222112122101112012000...

55/59=0,22101112012000220201012221121...

56/59=0,22112122101112012000220201012...

57/59=0,22200202121000110100121110210...

58/59=0,22211212210111201200022020101...

On remarque que le produit du nombre de périodes (2) et de leurs longueurs (29) est égal à 58 et donc au premier -1.