Il existe 2 périodes de 29 chiffres pour n/59 en base 4.

Pour toutes les fractions de n/59 en base 4, la période de 1/59 revient alors 29 fois (en orange)

1/59=0,00101112301321133131102130031...

2/59=0,00202231203302332322210320122...

3/59=0,00310010111230132113313110213...

4/59=0,01011123013211331311021300310...

5/59=0,01112301321133131102130031001...

6/59=0,01220020223120330233232221032...

7/59=0,01321133131102130031001011123...

8/59=0,02022312033023323222103201220...

9/59=0,02130031001011123013211331311...

10/59=0,02231203302332322210320122002...

11/59=0,02332322210320122002022312033...

12/59=0,03100101112301321133131102130...

13/59=0,03201220020223120330233232221...

14/59=0,03302332322210320122002022312...

15/59=0,10010111230132113313110213003...

16/59=0,10111230132113313110213003100...

17/59=0,10213003100101112301321133131...

18/59=0,10320122002022312033023323222...

19/59=0,11021300310010111230132113313...

20/59=0,11123013211331311021300310010...

21/59=0,11230132113313110213003100101...

22/59=0,11331311021300310010111230132...

23/59=0,12033023323222103201220020223...

24/59=0,12200202231203302332322210320...

25/59=0,12301321133131102130031001011...

26/59=0,13003100101112301321133131102...

27/59=0,13110213003100101112301321133...

28/59=0,13211331311021300310010111230...

29/59=0,13313110213003100101112301321...

30/59=0,20020223120330233232221032012...

31/59=0,20122002022312033023323222103...

32/59=0,20223120330233232221032012200...

33/59=0,20330233232221032012200202231...

34/59=0,21032012200202231203302332322...

35/59=0,21133131102130031001011123013...

36/59=0,21300310010111230132113313110...

37/59=0,22002022312033023323222103201...

38/59=0,22103201220020223120330233232...

39/59=0,22210320122002022312033023323...

40/59=0,22312033023323222103201220020...

41/59=0,23013211331311021300310010111...

42/59=0,23120330233232221032012200202...

43/59=0,23222103201220020223120330233...

44/59=0,23323222103201220020223120330...

45/59=0,30031001011123013211331311021...

46/59=0,30132113313110213003100101112...

47/59=0,30233232221032012200202231203...

48/59=0,31001011123013211331311021300...

49/59=0,31102130031001011123013211331...

50/59=0,31203302332322210320122002022...

51/59=0,31311021300310010111230132113...

52/59=0,32012200202231203302332322210...

53/59=0,32113313110213003100101112301...

54/59=0,32221032012200202231203302332...

55/59=0,32322210320122002022312033023...

56/59=0,33023323222103201220020223120...

57/59=0,33131102130031001011123013211...

58/59=0,33232221032012200202231203302...

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