Il existe 2 périodes de 99 chiffres pour n/199 en base 4.

Pour toutes les fractions de n/199 en base 4, la période de 1/199 revient alors 99 fois (en orange)

1/199=0,000110211103213203230231001212322202011302133201020002210222130330131211220030313110100232103330021...

2/199=0,000221022213033013121122003031311010023210333002100011021110321320323023100121232220201130213320102...

3/199=0,000331233322312223012013010310233212101113132203120013231333112311120300320212211330302022323310123...

4/199=0,001102111032132032302310012123222020113021332010200022102221303301312112200303131101002321033300210...

5/199=0,001212322202011302133201020002210222130330131211220030313110100232103330021000110211103213203230231...

6/199=0,001323133311231112030032021221133030202232331012300033123332231222301201301031023321210111313220312...

7/199=0,002100011021110321320323023100121232220201130213320102000221022213033013121122003031311010023210333...

8/199=0,002210222130330131211220030313110100232103330021000110211103213203230231001212322202011302133201020...

9/199=0,002321033300210001102111032132032302310012123222020113021332010200022102221303301312112200303131101...

10/199=0,003031311010023210333002100011021110321320323023100121232220201130213320102000221022213033013121122...

11/199=0,003202122113303020223233101230003312333223122230120130103102332121011131322031200132313331123111203...

12/199=0,003312333223122230120130103102332121011131322031200132313331123111203003202122113303020223233101230...

13/199=0,010023210333002100011021110321320323023100121232220201130213320102000221022213033013121122003031311...

14/199=0,010200022102221303301312112200303131101002321033300210001102111032132032302310012123222020113021332...

15/199=0,010310233212101113132203120013231333112311120300320212211330302022323310123000331233322312223012013...

16/199=0,011021110321320323023100121232220201130213320102000221022213033013121122003031311010023210333002100...

17/199=0,011131322031200132313331123111203003202122113303020223233101230003312333223122230120130103102332121...

18/199=0,011302133201020002210222130330131211220030313110100232103330021000110211103213203230231001212322202...

19/199=0,012013010310233212101113132203120013231333112311120300320212211330302022323310123000331233322312223...

20/199=0,012123222020113021332010200022102221303301312112200303131101002321033300210001102111032132032302310...

21/199=0,012300033123332231222301201301031023321210111313220312001323133311231112030032021221133030202232331...

22/199=0,013010310233212101113132203120013231333112311120300320212211330302022323310123000331233322312223012...

23/199=0,013121122003031311010023210333002100011021110321320323023100121232220201130213320102000221022213033...

24/199=0,013231333112311120300320212211330302022323310123000331233322312223012013010310233212101113132203120...

25/199=0,020002210222130330131211220030313110100232103330021000110211103213203230231001212322202011302133201...

26/199=0,020113021332010200022102221303301312112200303131101002321033300210001102111032132032302310012123222...

27/199=0,020223233101230003312333223122230120130103102332121011131322031200132313331123111203003202122113303...

28/199=0,021000110211103213203230231001212322202011302133201020002210222130330131211220030313110100232103330...

29/199=0,021110321320323023100121232220201130213320102000221022213033013121122003031311010023210333002100011...

30/199=0,021221133030202232331012300033123332231222301201301031023321210111313220312001323133311231112030032...

31/199=0,021332010200022102221303301312112200303131101002321033300210001102111032132032302310012123222020113...

32/199=0,022102221303301312112200303131101002321033300210001102111032132032302310012123222020113021332010200...

33/199=0,022213033013121122003031311010023210333002100011021110321320323023100121232220201130213320102000221...

34/199=0,022323310123000331233322312223012013010310233212101113132203120013231333112311120300320212211330302...

35/199=0,023100121232220201130213320102000221022213033013121122003031311010023210333002100011021110321320323...

36/199=0,023210333002100011021110321320323023100121232220201130213320102000221022213033013121122003031311010...

37/199=0,023321210111313220312001323133311231112030032021221133030202232331012300033123332231222301201301031...

38/199=0,030032021221133030202232331012300033123332231222301201301031023321210111313220312001323133311231112...

39/199=0,030202232331012300033123332231222301201301031023321210111313220312001323133311231112030032021221133...

40/199=0,030313110100232103330021000110211103213203230231001212322202011302133201020002210222130330131211220...

41/199=0,031023321210111313220312001323133311231112030032021221133030202232331012300033123332231222301201301...

42/199=0,031200132313331123111203003202122113303020223233101230003312333223122230120130103102332121011131322...

43/199=0,031311010023210333002100011021110321320323023100121232220201130213320102000221022213033013121122003...

44/199=0,032021221133030202232331012300033123332231222301201301031023321210111313220312001323133311231112030...

45/199=0,032132032302310012123222020113021332010200022102221303301312112200303131101002321033300210001102111...

46/199=0,032302310012123222020113021332010200022102221303301312112200303131101002321033300210001102111032132...

47/199=0,033013121122003031311010023210333002100011021110321320323023100121232220201130213320102000221022213...

48/199=0,033123332231222301201301031023321210111313220312001323133311231112030032021221133030202232331012300...

49/199=0,033300210001102111032132032302310012123222020113021332010200022102221303301312112200303131101002321...

50/199=0,100011021110321320323023100121232220201130213320102000221022213033013121122003031311010023210333002...

51/199=0,100121232220201130213320102000221022213033013121122003031311010023210333002100011021110321320323023...

52/199=0,100232103330021000110211103213203230231001212322202011302133201020002210222130330131211220030313110...

53/199=0,101002321033300210001102111032132032302310012123222020113021332010200022102221303301312112200303131...

54/199=0,101113132203120013231333112311120300320212211330302022323310123000331233322312223012013010310233212...

55/199=0,101230003312333223122230120130103102332121011131322031200132313331123111203003202122113303020223233...

56/199=0,102000221022213033013121122003031311010023210333002100011021110321320323023100121232220201130213320...

57/199=0,102111032132032302310012123222020113021332010200022102221303301312112200303131101002321033300210001...

58/199=0,102221303301312112200303131101002321033300210001102111032132032302310012123222020113021332010200022...

59/199=0,102332121011131322031200132313331123111203003202122113303020223233101230003312333223122230120130103...

60/199=0,103102332121011131322031200132313331123111203003202122113303020223233101230003312333223122230120130...

61/199=0,103213203230231001212322202011302133201020002210222130330131211220030313110100232103330021000110211...

62/199=0,103330021000110211103213203230231001212322202011302133201020002210222130330131211220030313110100232...

63/199=0,110100232103330021000110211103213203230231001212322202011302133201020002210222130330131211220030313...

64/199=0,110211103213203230231001212322202011302133201020002210222130330131211220030313110100232103330021000...

65/199=0,110321320323023100121232220201130213320102000221022213033013121122003031311010023210333002100011021...

66/199=0,111032132032302310012123222020113021332010200022102221303301312112200303131101002321033300210001102...

67/199=0,111203003202122113303020223233101230003312333223122230120130103102332121011131322031200132313331123...

68/199=0,111313220312001323133311231112030032021221133030202232331012300033123332231222301201301031023321210...

69/199=0,112030032021221133030202232331012300033123332231222301201301031023321210111313220312001323133311231...

70/199=0,112200303131101002321033300210001102111032132032302310012123222020113021332010200022102221303301312...

71/199=0,112311120300320212211330302022323310123000331233322312223012013010310233212101113132203120013231333...

72/199=0,113021332010200022102221303301312112200303131101002321033300210001102111032132032302310012123222020...

73/199=0,113132203120013231333112311120300320212211330302022323310123000331233322312223012013010310233212101...

74/199=0,113303020223233101230003312333223122230120130103102332121011131322031200132313331123111203003202122...

75/199=0,120013231333112311120300320212211330302022323310123000331233322312223012013010310233212101113132203...

76/199=0,120130103102332121011131322031200132313331123111203003202122113303020223233101230003312333223122230...

77/199=0,120300320212211330302022323310123000331233322312223012013010310233212101113132203120013231333112311...

78/199=0,121011131322031200132313331123111203003202122113303020223233101230003312333223122230120130103102332...

79/199=0,121122003031311010023210333002100011021110321320323023100121232220201130213320102000221022213033013...

80/199=0,121232220201130213320102000221022213033013121122003031311010023210333002100011021110321320323023100...

81/199=0,122003031311010023210333002100011021110321320323023100121232220201130213320102000221022213033013121...

82/199=0,122113303020223233101230003312333223122230120130103102332121011131322031200132313331123111203003202...

83/199=0,122230120130103102332121011131322031200132313331123111203003202122113303020223233101230003312333223...

84/199=0,123000331233322312223012013010310233212101113132203120013231333112311120300320212211330302022323310...

85/199=0,123111203003202122113303020223233101230003312333223122230120130103102332121011131322031200132313331...

86/199=0,123222020113021332010200022102221303301312112200303131101002321033300210001102111032132032302310012...

87/199=0,123332231222301201301031023321210111313220312001323133311231112030032021221133030202232331012300033...

88/199=0,130103102332121011131322031200132313331123111203003202122113303020223233101230003312333223122230120...

89/199=0,130213320102000221022213033013121122003031311010023210333002100011021110321320323023100121232220201...

90/199=0,130330131211220030313110100232103330021000110211103213203230231001212322202011302133201020002210222...

91/199=0,131101002321033300210001102111032132032302310012123222020113021332010200022102221303301312112200303...

92/199=0,131211220030313110100232103330021000110211103213203230231001212322202011302133201020002210222130330...

93/199=0,131322031200132313331123111203003202122113303020223233101230003312333223122230120130103102332121011...

94/199=0,132032302310012123222020113021332010200022102221303301312112200303131101002321033300210001102111032...

95/199=0,132203120013231333112311120300320212211330302022323310123000331233322312223012013010310233212101113...

96/199=0,132313331123111203003202122113303020223233101230003312333223122230120130103102332121011131322031200...

97/199=0,133030202232331012300033123332231222301201301031023321210111313220312001323133311231112030032021221...

98/199=0,133201020002210222130330131211220030313110100232103330021000110211103213203230231001212322202011302...

99/199=0,133311231112030032021221133030202232331012300033123332231222301201301031023321210111313220312001323...

100/199=0,200022102221303301312112200303131101002321033300210001102111032132032302310012123222020113021332010...

101/199=0,200132313331123111203003202122113303020223233101230003312333223122230120130103102332121011131322031...

102/199=0,200303131101002321033300210001102111032132032302310012123222020113021332010200022102221303301312112...

103/199=0,201020002210222130330131211220030313110100232103330021000110211103213203230231001212322202011302133...

104/199=0,201130213320102000221022213033013121122003031311010023210333002100011021110321320323023100121232220...

105/199=0,201301031023321210111313220312001323133311231112030032021221133030202232331012300033123332231222301...

106/199=0,202011302133201020002210222130330131211220030313110100232103330021000110211103213203230231001212322...

107/199=0,202122113303020223233101230003312333223122230120130103102332121011131322031200132313331123111203003...

108/199=0,202232331012300033123332231222301201301031023321210111313220312001323133311231112030032021221133030...

109/199=0,203003202122113303020223233101230003312333223122230120130103102332121011131322031200132313331123111...

110/199=0,203120013231333112311120300320212211330302022323310123000331233322312223012013010310233212101113132...

111/199=0,203230231001212322202011302133201020002210222130330131211220030313110100232103330021000110211103213...

112/199=0,210001102111032132032302310012123222020113021332010200022102221303301312112200303131101002321033300...

113/199=0,210111313220312001323133311231112030032021221133030202232331012300033123332231222301201301031023321...

114/199=0,210222130330131211220030313110100232103330021000110211103213203230231001212322202011302133201020002...

115/199=0,210333002100011021110321320323023100121232220201130213320102000221022213033013121122003031311010023...

116/199=0,211103213203230231001212322202011302133201020002210222130330131211220030313110100232103330021000110...

117/199=0,211220030313110100232103330021000110211103213203230231001212322202011302133201020002210222130330131...

118/199=0,211330302022323310123000331233322312223012013010310233212101113132203120013231333112311120300320212...

119/199=0,212101113132203120013231333112311120300320212211330302022323310123000331233322312223012013010310233...

120/199=0,212211330302022323310123000331233322312223012013010310233212101113132203120013231333112311120300320...

121/199=0,212322202011302133201020002210222130330131211220030313110100232103330021000110211103213203230231001...

122/199=0,213033013121122003031311010023210333002100011021110321320323023100121232220201130213320102000221022...

123/199=0,213203230231001212322202011302133201020002210222130330131211220030313110100232103330021000110211103...

124/199=0,213320102000221022213033013121122003031311010023210333002100011021110321320323023100121232220201130...

125/199=0,220030313110100232103330021000110211103213203230231001212322202011302133201020002210222130330131211...

126/199=0,220201130213320102000221022213033013121122003031311010023210333002100011021110321320323023100121232...

127/199=0,220312001323133311231112030032021221133030202232331012300033123332231222301201301031023321210111313...

128/199=0,221022213033013121122003031311010023210333002100011021110321320323023100121232220201130213320102000...

129/199=0,221133030202232331012300033123332231222301201301031023321210111313220312001323133311231112030032021...

130/199=0,221303301312112200303131101002321033300210001102111032132032302310012123222020113021332010200022102...

131/199=0,222020113021332010200022102221303301312112200303131101002321033300210001102111032132032302310012123...

132/199=0,222130330131211220030313110100232103330021000110211103213203230231001212322202011302133201020002210...

133/199=0,222301201301031023321210111313220312001323133311231112030032021221133030202232331012300033123332231...

134/199=0,223012013010310233212101113132203120013231333112311120300320212211330302022323310123000331233322312...

135/199=0,223122230120130103102332121011131322031200132313331123111203003202122113303020223233101230003312333...

136/199=0,223233101230003312333223122230120130103102332121011131322031200132313331123111203003202122113303020...

137/199=0,230003312333223122230120130103102332121011131322031200132313331123111203003202122113303020223233101...

138/199=0,230120130103102332121011131322031200132313331123111203003202122113303020223233101230003312333223122...

139/199=0,230231001212322202011302133201020002210222130330131211220030313110100232103330021000110211103213203...

140/199=0,231001212322202011302133201020002210222130330131211220030313110100232103330021000110211103213203230...

141/199=0,231112030032021221133030202232331012300033123332231222301201301031023321210111313220312001323133311...

142/199=0,231222301201301031023321210111313220312001323133311231112030032021221133030202232331012300033123332...

143/199=0,231333112311120300320212211330302022323310123000331233322312223012013010310233212101113132203120013...

144/199=0,232103330021000110211103213203230231001212322202011302133201020002210222130330131211220030313110100...

145/199=0,232220201130213320102000221022213033013121122003031311010023210333002100011021110321320323023100121...

146/199=0,232331012300033123332231222301201301031023321210111313220312001323133311231112030032021221133030202...

147/199=0,233101230003312333223122230120130103102332121011131322031200132313331123111203003202122113303020223...

148/199=0,233212101113132203120013231333112311120300320212211330302022323310123000331233322312223012013010310...

149/199=0,233322312223012013010310233212101113132203120013231333112311120300320212211330302022323310123000331...

150/199=0,300033123332231222301201301031023321210111313220312001323133311231112030032021221133030202232331012...

151/199=0,300210001102111032132032302310012123222020113021332010200022102221303301312112200303131101002321033...

152/199=0,300320212211330302022323310123000331233322312223012013010310233212101113132203120013231333112311120...

153/199=0,301031023321210111313220312001323133311231112030032021221133030202232331012300033123332231222301201...

154/199=0,301201301031023321210111313220312001323133311231112030032021221133030202232331012300033123332231222...

155/199=0,301312112200303131101002321033300210001102111032132032302310012123222020113021332010200022102221303...

156/199=0,302022323310123000331233322312223012013010310233212101113132203120013231333112311120300320212211330...

157/199=0,302133201020002210222130330131211220030313110100232103330021000110211103213203230231001212322202011...

158/199=0,302310012123222020113021332010200022102221303301312112200303131101002321033300210001102111032132032...

159/199=0,303020223233101230003312333223122230120130103102332121011131322031200132313331123111203003202122113...

160/199=0,303131101002321033300210001102111032132032302310012123222020113021332010200022102221303301312112200...

161/199=0,303301312112200303131101002321033300210001102111032132032302310012123222020113021332010200022102221...

162/199=0,310012123222020113021332010200022102221303301312112200303131101002321033300210001102111032132032302...

163/199=0,310123000331233322312223012013010310233212101113132203120013231333112311120300320212211330302022323...

164/199=0,310233212101113132203120013231333112311120300320212211330302022323310123000331233322312223012013010...

165/199=0,311010023210333002100011021110321320323023100121232220201130213320102000221022213033013121122003031...

166/199=0,311120300320212211330302022323310123000331233322312223012013010310233212101113132203120013231333112...

167/199=0,311231112030032021221133030202232331012300033123332231222301201301031023321210111313220312001323133...

168/199=0,312001323133311231112030032021221133030202232331012300033123332231222301201301031023321210111313220...

169/199=0,312112200303131101002321033300210001102111032132032302310012123222020113021332010200022102221303301...

170/199=0,312223012013010310233212101113132203120013231333112311120300320212211330302022323310123000331233322...

171/199=0,312333223122230120130103102332121011131322031200132313331123111203003202122113303020223233101230003...

172/199=0,313110100232103330021000110211103213203230231001212322202011302133201020002210222130330131211220030...

173/199=0,313220312001323133311231112030032021221133030202232331012300033123332231222301201301031023321210111...

174/199=0,313331123111203003202122113303020223233101230003312333223122230120130103102332121011131322031200132...

175/199=0,320102000221022213033013121122003031311010023210333002100011021110321320323023100121232220201130213...

176/199=0,320212211330302022323310123000331233322312223012013010310233212101113132203120013231333112311120300...

177/199=0,320323023100121232220201130213320102000221022213033013121122003031311010023210333002100011021110321...

178/199=0,321033300210001102111032132032302310012123222020113021332010200022102221303301312112200303131101002...

179/199=0,321210111313220312001323133311231112030032021221133030202232331012300033123332231222301201301031023...

180/199=0,321320323023100121232220201130213320102000221022213033013121122003031311010023210333002100011021110...

181/199=0,322031200132313331123111203003202122113303020223233101230003312333223122230120130103102332121011131...

182/199=0,322202011302133201020002210222130330131211220030313110100232103330021000110211103213203230231001212...

183/199=0,322312223012013010310233212101113132203120013231333112311120300320212211330302022323310123000331233...

184/199=0,323023100121232220201130213320102000221022213033013121122003031311010023210333002100011021110321320...

185/199=0,323133311231112030032021221133030202232331012300033123332231222301201301031023321210111313220312001...

186/199=0,323310123000331233322312223012013010310233212101113132203120013231333112311120300320212211330302022...

187/199=0,330021000110211103213203230231001212322202011302133201020002210222130330131211220030313110100232103...

188/199=0,330131211220030313110100232103330021000110211103213203230231001212322202011302133201020002210222130...

189/199=0,330302022323310123000331233322312223012013010310233212101113132203120013231333112311120300320212211...

190/199=0,331012300033123332231222301201301031023321210111313220312001323133311231112030032021221133030202232...

191/199=0,331123111203003202122113303020223233101230003312333223122230120130103102332121011131322031200132313...

192/199=0,331233322312223012013010310233212101113132203120013231333112311120300320212211330302022323310123000...

193/199=0,332010200022102221303301312112200303131101002321033300210001102111032132032302310012123222020113021...

194/199=0,332121011131322031200132313331123111203003202122113303020223233101230003312333223122230120130103102...

195/199=0,332231222301201301031023321210111313220312001323133311231112030032021221133030202232331012300033123...

196/199=0,333002100011021110321320323023100121232220201130213320102000221022213033013121122003031311010023210...

197/199=0,333112311120300320212211330302022323310123000331233322312223012013010310233212101113132203120013231...

198/199=0,333223122230120130103102332121011131322031200132313331123111203003202122113303020223233101230003312...

On remarque que le produit du nombre de périodes (2) et de leurs longueurs (99) est égal à 198 et donc au premier -1.