Il existe 8 périodes de 14 chiffres pour n/113 en base 7.

Pour toutes les fractions de n/113 en base 7, la période de 1/113 revient alors 14 fois (en orange)

1/113=0,00301506636516...

2/113=0,00603316606335...

3/113=0,01205126546154...

4/113=0,01506636516003...

5/113=0,02111446455522...

6/113=0,02413256425341...

7/113=0,03015066365160...

8/113=0,03316606335006...

9/113=0,03621416304525...

10/113=0,04223226244344...

11/113=0,04525036214163...

12/113=0,05126546154012...

13/113=0,05431356123531...

14/113=0,06033166063350...

15/113=0,06335006033166...

16/113=0,06636516003015...

17/113=0,10241325642534...

18/113=0,10543135612353...

19/113=0,11144645552202...

20/113=0,11446455522021...

21/113=0,12051265461540...

22/113=0,12353105431356...

23/113=0,12654615401205...

24/113=0,13256425341024...

25/113=0,13561235310543...

26/113=0,14163045250362...

27/113=0,14464555220211...

28/113=0,15066365160030...

29/113=0,15401205126546...

30/113=0,16003015066365...

31/113=0,16304525036214...

32/113=0,16606335006033...

33/113=0,20211144645552...

34/113=0,20512654615401...

35/113=0,21114464555220...

36/113=0,21416304525036...

37/113=0,22021114464555...

38/113=0,22322624434404...

39/113=0,22624434404223...

40/113=0,23226244344042...

41/113=0,23531054313561...

42/113=0,24132564253410...

43/113=0,24434404223226...

44/113=0,25036214163045...

45/113=0,25341024132564...

46/113=0,25642534102413...

47/113=0,26244344042232...

48/113=0,26546154012051...

49/113=0,30150663651600...

50/113=0,30452503621416...

51/113=0,31054313561235...

52/113=0,31356123531054...

53/113=0,31660633500603...

54/113=0,32262443440422...

55/113=0,32564253410241...

56/113=0,33166063350060...

57/113=0,33500603316606...

58/113=0,34102413256425...

59/113=0,34404223226244...

60/113=0,35006033166063...

61/113=0,35310543135612...

62/113=0,35612353105431...

63/113=0,36214163045250...

64/113=0,36516003015066...

65/113=0,40120512654615...

66/113=0,40422322624434...

67/113=0,41024132564253...

68/113=0,41325642534102...

69/113=0,41630452503621...

70/113=0,42232262443440...

71/113=0,42534102413256...

72/113=0,43135612353105...

73/113=0,43440422322624...

74/113=0,44042232262443...

75/113=0,44344042232262...

76/113=0,44645552202111...

77/113=0,45250362141630...

78/113=0,45552202111446...

79/113=0,46154012051265...

80/113=0,46455522021114...

81/113=0,50060331660633...

82/113=0,50362141630452...

83/113=0,50663651600301...

84/113=0,51265461540120...

85/113=0,51600301506636...

86/113=0,52202111446455...

87/113=0,52503621416304...

88/113=0,53105431356123...

89/113=0,53410241325642...

90/113=0,54012051265461...

91/113=0,54313561235310...

92/113=0,54615401205126...

93/113=0,55220211144645...

94/113=0,55522021114464...

95/113=0,56123531054313...

96/113=0,56425341024132...

97/113=0,60030150663651...

98/113=0,60331660633500...

99/113=0,60633500603316...

100/113=0,61235310543135...

101/113=0,61540120512654...

102/113=0,62141630452503...

103/113=0,62443440422322...

104/113=0,63045250362141...

105/113=0,63350060331660...

106/113=0,63651600301506...

107/113=0,64253410241325...

108/113=0,64555220211144...

109/113=0,65160030150663...

110/113=0,65461540120512...

111/113=0,66063350060331...

112/113=0,66365160030150...

On remarque que le produit du nombre de périodes (8) et de leurs longueurs (14) est égal à 112 et donc au premier -1.