Il existe 2 périodes de 105 chiffres pour n/211 en base 4.

Pour toutes les fractions de n/211 en base 4, la période de 1/211 revient alors 105 fois (en orange)

1/211=0,000103122120313303313200131013211103231303100223223113331210230321001201003330113003111202101211322202211...

2/211=0,000212310301233213233000322033022213123212201113112233323021121302003002013320232012223010203023311011022...

3/211=0,000322033022213123212201113112233323021121302003002013320232012223010203023311011022000212310301233213233...

4/211=0,001031221203133033132001310132111032313031002232231133312102303210012010033301130031112021012113222022110...

5/211=0,001201003330113003111202101211322202211000103122120313303313200131013211103231303100223223113331210230321...

6/211=0,001310132111032313031002232231133312102303210012010033301130031112021012113222022110001031221203133033132...

7/211=0,002013320232012223010203023311011022000212310301233213233000322033022213123212201113112233323021121302003...

8/211=0,002123103012332132330003220330222131232122011131122333230211213020030020133202320122230102030233110110220...

9/211=0,002232231133312102303210012010033301130031112021012113222022110001031221203133033132001310132111032313031...

10/211=0,003002013320232012223010203023311011022000212310301233213233000322033022213123212201113112233323021121302...

11/211=0,003111202101211322202211000103122120313303313200131013211103231303100223223113331210230321001201003330113...

12/211=0,003220330222131232122011131122333230211213020030020133202320122230102030233110110220002123103012332132330...

13/211=0,003330113003111202101211322202211000103122120313303313200131013211103231303100223223113331210230321001201...

14/211=0,010033301130031112021012113222022110001031221203133033132001310132111032313031002232231133312102303210012...

15/211=0,010203023311011022000212310301233213233000322033022213123212201113112233323021121302003002013320232012223...

16/211=0,010312212031330331320013101321110323130310022322311333121023032100120100333011300311120210121132220221100...

17/211=0,011022000212310301233213233000322033022213123212201113112233323021121302003002013320232012223010203023311...

18/211=0,011131122333230211213020030020133202320122230102030233110110220002123103012332132330003220330222131232122...

19/211=0,011300311120210121132220221100010312212031330331320013101321110323130310022322311333121023032100120100333...

20/211=0,012010033301130031112021012113222022110001031221203133033132001310132111032313031002232231133312102303210...

21/211=0,012113222022110001031221203133033132001310132111032313031002232231133312102303210012010033301130031112021...

22/211=0,012223010203023311011022000212310301233213233000322033022213123212201113112233323021121302003002013320232...

23/211=0,012332132330003220330222131232122011131122333230211213020030020133202320122230102030233110110220002123103...

24/211=0,013101321110323130310022322311333121023032100120100333011300311120210121132220221100010312212031330331320...

25/211=0,013211103231303100223223113331210230321001201003330113003111202101211322202211000103122120313303313200131...

26/211=0,013320232012223010203023311011022000212310301233213233000322033022213123212201113112233323021121302003002...

27/211=0,020030020133202320122230102030233110110220002123103012332132330003220330222131232122011131122333230211213...

28/211=0,020133202320122230102030233110110220002123103012332132330003220330222131232122011131122333230211213020030...

29/211=0,020302331101102200021231030123321323300032203302221312321220111311223332302112130200300201332023201222301...

30/211=0,021012113222022110001031221203133033132001310132111032313031002232231133312102303210012010033301130031112...

31/211=0,021121302003002013320232012223010203023311011022000212310301233213233000322033022213123212201113112233323...

32/211=0,021231030123321323300032203302221312321220111311223332302112130200300201332023201222301020302331101102200...

33/211=0,022000212310301233213233000322033022213123212201113112233323021121302003002013320232012223010203023311011...

34/211=0,022110001031221203133033132001310132111032313031002232231133312102303210012010033301130031112021012113222...

35/211=0,022213123212201113112233323021121302003002013320232012223010203023311011022000212310301233213233000322033...

36/211=0,022322311333121023032100120100333011300311120210121132220221100010312212031330331320013101321110323130310...

37/211=0,023032100120100333011300311120210121132220221100010312212031330331320013101321110323130310022322311333121...

38/211=0,023201222301020302331101102200021231030123321323300032203302221312321220111311223332302112130200300201332...

39/211=0,023311011022000212310301233213233000322033022213123212201113112233323021121302003002013320232012223010203...

40/211=0,030020133202320122230102030233110110220002123103012332132330003220330222131232122011131122333230211213020...

41/211=0,030123321323300032203302221312321220111311223332302112130200300201332023201222301020302331101102200021231...

42/211=0,030233110110220002123103012332132330003220330222131232122011131122333230211213020030020133202320122230102...

43/211=0,031002232231133312102303210012010033301130031112021012113222022110001031221203133033132001310132111032313...

44/211=0,031112021012113222022110001031221203133033132001310132111032313031002232231133312102303210012010033301130...

45/211=0,031221203133033132001310132111032313031002232231133312102303210012010033301130031112021012113222022110001...

46/211=0,031330331320013101321110323130310022322311333121023032100120100333011300311120210121132220221100010312212...

47/211=0,032100120100333011300311120210121132220221100010312212031330331320013101321110323130310022322311333121023...

48/211=0,032203302221312321220111311223332302112130200300201332023201222301020302331101102200021231030123321323300...

49/211=0,032313031002232231133312102303210012010033301130031112021012113222022110001031221203133033132001310132111...

50/211=0,033022213123212201113112233323021121302003002013320232012223010203023311011022000212310301233213233000322...

51/211=0,033132001310132111032313031002232231133312102303210012010033301130031112021012113222022110001031221203133...

52/211=0,033301130031112021012113222022110001031221203133033132001310132111032313031002232231133312102303210012010...

53/211=0,100010312212031330331320013101321110323130310022322311333121023032100120100333011300311120210121132220221...

54/211=0,100120100333011300311120210121132220221100010312212031330331320013101321110323130310022322311333121023032...

55/211=0,100223223113331210230321001201003330113003111202101211322202211000103122120313303313200131013211103231303...

56/211=0,100333011300311120210121132220221100010312212031330331320013101321110323130310022322311333121023032100120...

57/211=0,101102200021231030123321323300032203302221312321220111311223332302112130200300201332023201222301020302331...

58/211=0,101211322202211000103122120313303313200131013211103231303100223223113331210230321001201003330113003111202...

59/211=0,101321110323130310022322311333121023032100120100333011300311120210121132220221100010312212031330331320013...

60/211=0,102030233110110220002123103012332132330003220330222131232122011131122333230211213020030020133202320122230...

61/211=0,102200021231030123321323300032203302221312321220111311223332302112130200300201332023201222301020302331101...

62/211=0,102303210012010033301130031112021012113222022110001031221203133033132001310132111032313031002232231133312...

63/211=0,103012332132330003220330222131232122011131122333230211213020030020133202320122230102030233110110220002123...

64/211=0,103122120313303313200131013211103231303100223223113331210230321001201003330113003111202101211322202211000...

65/211=0,103231303100223223113331210230321001201003330113003111202101211322202211000103122120313303313200131013211...

66/211=0,110001031221203133033132001310132111032313031002232231133312102303210012010033301130031112021012113222022...

67/211=0,110110220002123103012332132330003220330222131232122011131122333230211213020030020133202320122230102030233...

68/211=0,110220002123103012332132330003220330222131232122011131122333230211213020030020133202320122230102030233110...

69/211=0,110323130310022322311333121023032100120100333011300311120210121132220221100010312212031330331320013101321...

70/211=0,111032313031002232231133312102303210012010033301130031112021012113222022110001031221203133033132001310132...

71/211=0,111202101211322202211000103122120313303313200131013211103231303100223223113331210230321001201003330113003...

72/211=0,111311223332302112130200300201332023201222301020302331101102200021231030123321323300032203302221312321220...

73/211=0,112021012113222022110001031221203133033132001310132111032313031002232231133312102303210012010033301130031...

74/211=0,112130200300201332023201222301020302331101102200021231030123321323300032203302221312321220111311223332302...

75/211=0,112233323021121302003002013320232012223010203023311011022000212310301233213233000322033022213123212201113...

76/211=0,113003111202101211322202211000103122120313303313200131013211103231303100223223113331210230321001201003330...

77/211=0,113112233323021121302003002013320232012223010203023311011022000212310301233213233000322033022213123212201...

78/211=0,113222022110001031221203133033132001310132111032313031002232231133312102303210012010033301130031112021012...

79/211=0,113331210230321001201003330113003111202101211322202211000103122120313303313200131013211103231303100223223...

80/211=0,120100333011300311120210121132220221100010312212031330331320013101321110323130310022322311333121023032100...

81/211=0,120210121132220221100010312212031330331320013101321110323130310022322311333121023032100120100333011300311...

82/211=0,120313303313200131013211103231303100223223113331210230321001201003330113003111202101211322202211000103122...

83/211=0,121023032100120100333011300311120210121132220221100010312212031330331320013101321110323130310022322311333...

84/211=0,121132220221100010312212031330331320013101321110323130310022322311333121023032100120100333011300311120210...

85/211=0,121302003002013320232012223010203023311011022000212310301233213233000322033022213123212201113112233323021...

86/211=0,122011131122333230211213020030020133202320122230102030233110110220002123103012332132330003220330222131232...

87/211=0,122120313303313200131013211103231303100223223113331210230321001201003330113003111202101211322202211000103...

88/211=0,122230102030233110110220002123103012332132330003220330222131232122011131122333230211213020030020133202320...

89/211=0,122333230211213020030020133202320122230102030233110110220002123103012332132330003220330222131232122011131...

90/211=0,123103012332132330003220330222131232122011131122333230211213020030020133202320122230102030233110110220002...

91/211=0,123212201113112233323021121302003002013320232012223010203023311011022000212310301233213233000322033022213...

92/211=0,123321323300032203302221312321220111311223332302112130200300201332023201222301020302331101102200021231030...

93/211=0,130031112021012113222022110001031221203133033132001310132111032313031002232231133312102303210012010033301...

94/211=0,130200300201332023201222301020302331101102200021231030123321323300032203302221312321220111311223332302112...

95/211=0,130310022322311333121023032100120100333011300311120210121132220221100010312212031330331320013101321110323...

96/211=0,131013211103231303100223223113331210230321001201003330113003111202101211322202211000103122120313303313200...

97/211=0,131122333230211213020030020133202320122230102030233110110220002123103012332132330003220330222131232122011...

98/211=0,131232122011131122333230211213020030020133202320122230102030233110110220002123103012332132330003220330222...

99/211=0,132001310132111032313031002232231133312102303210012010033301130031112021012113222022110001031221203133033...

100/211=0,132111032313031002232231133312102303210012010033301130031112021012113222022110001031221203133033132001310...

101/211=0,132220221100010312212031330331320013101321110323130310022322311333121023032100120100333011300311120210121...

102/211=0,132330003220330222131232122011131122333230211213020030020133202320122230102030233110110220002123103012332...

103/211=0,133033132001310132111032313031002232231133312102303210012010033301130031112021012113222022110001031221203...

104/211=0,133202320122230102030233110110220002123103012332132330003220330222131232122011131122333230211213020030020...

105/211=0,133312102303210012010033301130031112021012113222022110001031221203133033132001310132111032313031002232231...

106/211=0,200021231030123321323300032203302221312321220111311223332302112130200300201332023201222301020302331101102...

107/211=0,200131013211103231303100223223113331210230321001201003330113003111202101211322202211000103122120313303313...

108/211=0,200300201332023201222301020302331101102200021231030123321323300032203302221312321220111311223332302112130...

109/211=0,201003330113003111202101211322202211000103122120313303313200131013211103231303100223223113331210230321001...

110/211=0,201113112233323021121302003002013320232012223010203023311011022000212310301233213233000322033022213123212...

111/211=0,201222301020302331101102200021231030123321323300032203302221312321220111311223332302112130200300201332023...

112/211=0,201332023201222301020302331101102200021231030123321323300032203302221312321220111311223332302112130200300...

113/211=0,202101211322202211000103122120313303313200131013211103231303100223223113331210230321001201003330113003111...

114/211=0,202211000103122120313303313200131013211103231303100223223113331210230321001201003330113003111202101211322...

115/211=0,202320122230102030233110110220002123103012332132330003220330222131232122011131122333230211213020030020133...

116/211=0,203023311011022000212310301233213233000322033022213123212201113112233323021121302003002013320232012223010...

117/211=0,203133033132001310132111032313031002232231133312102303210012010033301130031112021012113222022110001031221...

118/211=0,203302221312321220111311223332302112130200300201332023201222301020302331101102200021231030123321323300032...

119/211=0,210012010033301130031112021012113222022110001031221203133033132001310132111032313031002232231133312102303...

120/211=0,210121132220221100010312212031330331320013101321110323130310022322311333121023032100120100333011300311120...

121/211=0,210230321001201003330113003111202101211322202211000103122120313303313200131013211103231303100223223113331...

122/211=0,211000103122120313303313200131013211103231303100223223113331210230321001201003330113003111202101211322202...

123/211=0,211103231303100223223113331210230321001201003330113003111202101211322202211000103122120313303313200131013...

124/211=0,211213020030020133202320122230102030233110110220002123103012332132330003220330222131232122011131122333230...

125/211=0,211322202211000103122120313303313200131013211103231303100223223113331210230321001201003330113003111202101...

126/211=0,212031330331320013101321110323130310022322311333121023032100120100333011300311120210121132220221100010312...

127/211=0,212201113112233323021121302003002013320232012223010203023311011022000212310301233213233000322033022213123...

128/211=0,212310301233213233000322033022213123212201113112233323021121302003002013320232012223010203023311011022000...

129/211=0,213020030020133202320122230102030233110110220002123103012332132330003220330222131232122011131122333230211...

130/211=0,213123212201113112233323021121302003002013320232012223010203023311011022000212310301233213233000322033022...

131/211=0,213233000322033022213123212201113112233323021121302003002013320232012223010203023311011022000212310301233...

132/211=0,220002123103012332132330003220330222131232122011131122333230211213020030020133202320122230102030233110110...

133/211=0,220111311223332302112130200300201332023201222301020302331101102200021231030123321323300032203302221312321...

134/211=0,220221100010312212031330331320013101321110323130310022322311333121023032100120100333011300311120210121132...

135/211=0,220330222131232122011131122333230211213020030020133202320122230102030233110110220002123103012332132330003...

136/211=0,221100010312212031330331320013101321110323130310022322311333121023032100120100333011300311120210121132220...

137/211=0,221203133033132001310132111032313031002232231133312102303210012010033301130031112021012113222022110001031...

138/211=0,221312321220111311223332302112130200300201332023201222301020302331101102200021231030123321323300032203302...

139/211=0,222022110001031221203133033132001310132111032313031002232231133312102303210012010033301130031112021012113...

140/211=0,222131232122011131122333230211213020030020133202320122230102030233110110220002123103012332132330003220330...

141/211=0,222301020302331101102200021231030123321323300032203302221312321220111311223332302112130200300201332023201...

142/211=0,223010203023311011022000212310301233213233000322033022213123212201113112233323021121302003002013320232012...

143/211=0,223113331210230321001201003330113003111202101211322202211000103122120313303313200131013211103231303100223...

144/211=0,223223113331210230321001201003330113003111202101211322202211000103122120313303313200131013211103231303100...

145/211=0,223332302112130200300201332023201222301020302331101102200021231030123321323300032203302221312321220111311...

146/211=0,230102030233110110220002123103012332132330003220330222131232122011131122333230211213020030020133202320122...

147/211=0,230211213020030020133202320122230102030233110110220002123103012332132330003220330222131232122011131122333...

148/211=0,230321001201003330113003111202101211322202211000103122120313303313200131013211103231303100223223113331210...

149/211=0,231030123321323300032203302221312321220111311223332302112130200300201332023201222301020302331101102200021...

150/211=0,231133312102303210012010033301130031112021012113222022110001031221203133033132001310132111032313031002232...

151/211=0,231303100223223113331210230321001201003330113003111202101211322202211000103122120313303313200131013211103...

152/211=0,232012223010203023311011022000212310301233213233000322033022213123212201113112233323021121302003002013320...

153/211=0,232122011131122333230211213020030020133202320122230102030233110110220002123103012332132330003220330222131...

154/211=0,232231133312102303210012010033301130031112021012113222022110001031221203133033132001310132111032313031002...

155/211=0,233000322033022213123212201113112233323021121302003002013320232012223010203023311011022000212310301233213...

156/211=0,233110110220002123103012332132330003220330222131232122011131122333230211213020030020133202320122230102030...

157/211=0,233213233000322033022213123212201113112233323021121302003002013320232012223010203023311011022000212310301...

158/211=0,233323021121302003002013320232012223010203023311011022000212310301233213233000322033022213123212201113112...

159/211=0,300032203302221312321220111311223332302112130200300201332023201222301020302331101102200021231030123321323...

160/211=0,300201332023201222301020302331101102200021231030123321323300032203302221312321220111311223332302112130200...

161/211=0,300311120210121132220221100010312212031330331320013101321110323130310022322311333121023032100120100333011...

162/211=0,301020302331101102200021231030123321323300032203302221312321220111311223332302112130200300201332023201222...

163/211=0,301130031112021012113222022110001031221203133033132001310132111032313031002232231133312102303210012010033...

164/211=0,301233213233000322033022213123212201113112233323021121302003002013320232012223010203023311011022000212310...

165/211=0,302003002013320232012223010203023311011022000212310301233213233000322033022213123212201113112233323021121...

166/211=0,302112130200300201332023201222301020302331101102200021231030123321323300032203302221312321220111311223332...

167/211=0,302221312321220111311223332302112130200300201332023201222301020302331101102200021231030123321323300032203...

168/211=0,302331101102200021231030123321323300032203302221312321220111311223332302112130200300201332023201222301020...

169/211=0,303100223223113331210230321001201003330113003111202101211322202211000103122120313303313200131013211103231...

170/211=0,303210012010033301130031112021012113222022110001031221203133033132001310132111032313031002232231133312102...

171/211=0,303313200131013211103231303100223223113331210230321001201003330113003111202101211322202211000103122120313...

172/211=0,310022322311333121023032100120100333011300311120210121132220221100010312212031330331320013101321110323130...

173/211=0,310132111032313031002232231133312102303210012010033301130031112021012113222022110001031221203133033132001...

174/211=0,310301233213233000322033022213123212201113112233323021121302003002013320232012223010203023311011022000212...

175/211=0,311011022000212310301233213233000322033022213123212201113112233323021121302003002013320232012223010203023...

176/211=0,311120210121132220221100010312212031330331320013101321110323130310022322311333121023032100120100333011300...

177/211=0,311223332302112130200300201332023201222301020302331101102200021231030123321323300032203302221312321220111...

178/211=0,311333121023032100120100333011300311120210121132220221100010312212031330331320013101321110323130310022322...

179/211=0,312102303210012010033301130031112021012113222022110001031221203133033132001310132111032313031002232231133...

180/211=0,312212031330331320013101321110323130310022322311333121023032100120100333011300311120210121132220221100010...

181/211=0,312321220111311223332302112130200300201332023201222301020302331101102200021231030123321323300032203302221...

182/211=0,313031002232231133312102303210012010033301130031112021012113222022110001031221203133033132001310132111032...

183/211=0,313200131013211103231303100223223113331210230321001201003330113003111202101211322202211000103122120313303...

184/211=0,313303313200131013211103231303100223223113331210230321001201003330113003111202101211322202211000103122120...

185/211=0,320013101321110323130310022322311333121023032100120100333011300311120210121132220221100010312212031330331...

186/211=0,320122230102030233110110220002123103012332132330003220330222131232122011131122333230211213020030020133202...

187/211=0,320232012223010203023311011022000212310301233213233000322033022213123212201113112233323021121302003002013...

188/211=0,321001201003330113003111202101211322202211000103122120313303313200131013211103231303100223223113331210230...

189/211=0,321110323130310022322311333121023032100120100333011300311120210121132220221100010312212031330331320013101...

190/211=0,321220111311223332302112130200300201332023201222301020302331101102200021231030123321323300032203302221312...

191/211=0,321323300032203302221312321220111311223332302112130200300201332023201222301020302331101102200021231030123...

192/211=0,322033022213123212201113112233323021121302003002013320232012223010203023311011022000212310301233213233000...

193/211=0,322202211000103122120313303313200131013211103231303100223223113331210230321001201003330113003111202101211...

194/211=0,322311333121023032100120100333011300311120210121132220221100010312212031330331320013101321110323130310022...

195/211=0,323021121302003002013320232012223010203023311011022000212310301233213233000322033022213123212201113112233...

196/211=0,323130310022322311333121023032100120100333011300311120210121132220221100010312212031330331320013101321110...

197/211=0,323300032203302221312321220111311223332302112130200300201332023201222301020302331101102200021231030123321...

198/211=0,330003220330222131232122011131122333230211213020030020133202320122230102030233110110220002123103012332132...

199/211=0,330113003111202101211322202211000103122120313303313200131013211103231303100223223113331210230321001201003...

200/211=0,330222131232122011131122333230211213020030020133202320122230102030233110110220002123103012332132330003220...

201/211=0,330331320013101321110323130310022322311333121023032100120100333011300311120210121132220221100010312212031...

202/211=0,331101102200021231030123321323300032203302221312321220111311223332302112130200300201332023201222301020302...

203/211=0,331210230321001201003330113003111202101211322202211000103122120313303313200131013211103231303100223223113...

204/211=0,331320013101321110323130310022322311333121023032100120100333011300311120210121132220221100010312212031330...

205/211=0,332023201222301020302331101102200021231030123321323300032203302221312321220111311223332302112130200300201...

206/211=0,332132330003220330222131232122011131122333230211213020030020133202320122230102030233110110220002123103012...

207/211=0,332302112130200300201332023201222301020302331101102200021231030123321323300032203302221312321220111311223...

208/211=0,333011300311120210121132220221100010312212031330331320013101321110323130310022322311333121023032100120100...

209/211=0,333121023032100120100333011300311120210121132220221100010312212031330331320013101321110323130310022322311...

210/211=0,333230211213020030020133202320122230102030233110110220002123103012332132330003220330222131232122011131122...

On remarque que le produit du nombre de périodes (2) et de leurs longueurs (105) est égal à 210 et donc au premier -1.