Il existe 2 périodes de 20 chiffres pour n/41 en base 2.

Pour toutes les fractions de n/41 en base 2, la période de 1/41 revient alors 20 fois (en orange)

1/41=0,00000110001111100111...

2/41=0,00001100011111001110...

3/41=0,00010010101110110101...

4/41=0,00011000111110011100...

5/41=0,00011111001110000011...

6/41=0,00100101011101101010...

7/41=0,00101011101101010001...

8/41=0,00110001111100111000...

9/41=0,00111000001100011111...

10/41=0,00111110011100000110...

11/41=0,01000100101011101101...

12/41=0,01001010111011010100...

13/41=0,01010001001010111011...

14/41=0,01010111011010100010...

15/41=0,01011101101010001001...

16/41=0,01100011111001110000...

17/41=0,01101010001001010111...

18/41=0,01110000011000111110...

19/41=0,01110110101000100101...

20/41=0,01111100111000001100...

21/41=0,10000011000111110011...

22/41=0,10001001010111011010...

23/41=0,10001111100111000001...

24/41=0,10010101110110101000...

25/41=0,10011100000110001111...

26/41=0,10100010010101110110...

27/41=0,10101000100101011101...

28/41=0,10101110110101000100...

29/41=0,10110101000100101011...

30/41=0,10111011010100010010...

31/41=0,11000001100011111001...

32/41=0,11000111110011100000...

33/41=0,11001110000011000111...

34/41=0,11010100010010101110...

35/41=0,11011010100010010101...

36/41=0,11100000110001111100...

37/41=0,11100111000001100011...

38/41=0,11101101010001001010...

39/41=0,11110011100000110001...

40/41=0,11111001110000011000...

On remarque que le produit du nombre de périodes (2) et de leurs longueurs (20) est égal à 40 et donc au premier -1.