Il existe 2 périodes de 18 chiffres pour n/37 en base 3.

Pour toutes les fractions de n/37 en base 3, la période de 1/37 revient alors 18 fois (en orange)

1/37=0,000201200222021022...

2/37=0,001110101221112121...

3/37=0,002012002220210220...

4/37=0,002220210220002012...

5/37=0,010122111212100111...

6/37=0,011101012211121210...

7/37=0,012002220210220002...

8/37=0,012211121210011101...

9/37=0,020120022202102200...

10/37=0,021022000201200222...

11/37=0,022000201200222021...

12/37=0,022202102200020120...

13/37=0,100111010122111212...

14/37=0,101012211121210011...

15/37=0,101221112121001110...

16/37=0,102200020120022202...

17/37=0,110101221112121001...

18/37=0,111010122111212100...

19/37=0,111212100111010122...

20/37=0,112121001110101221...

21/37=0,120022202102200020...

22/37=0,121001110101221112...

23/37=0,121210011101012211...

24/37=0,122111212100111010...

25/37=0,200020120022202102...

26/37=0,200222021022000201...

27/37=0,201200222021022000...

28/37=0,202102200020120022...

29/37=0,210011101012211121...

30/37=0,210220002012002220...

31/37=0,211121210011101012...

32/37=0,212100111010122111...

33/37=0,220002012002220210...

34/37=0,220210220002012002...

35/37=0,221112121001110101...

36/37=0,222021022000201200...

On remarque que le produit du nombre de périodes (2) et de leurs longueurs (18) est égal à 36 et donc au premier -1.