Il existe 16 périodes de 16 chiffres pour n/257 en base 2.

Pour toutes les fractions de n/257 en base 2, la période de 1/257 revient alors 16 fois (en orange)

1/257=0,0000000011111111...

2/257=0,0000000111111110...

3/257=0,0000001011111101...

4/257=0,0000001111111100...

5/257=0,0000010011111011...

6/257=0,0000010111111010...

7/257=0,0000011011111001...

8/257=0,0000011111111000...

9/257=0,0000100011110111...

10/257=0,0000100111110110...

11/257=0,0000101011110101...

12/257=0,0000101111110100...

13/257=0,0000110011110011...

14/257=0,0000110111110010...

15/257=0,0000111011110001...

16/257=0,0000111111110000...

17/257=0,0001000011101111...

18/257=0,0001000111101110...

19/257=0,0001001011101101...

20/257=0,0001001111101100...

21/257=0,0001010011101011...

22/257=0,0001010111101010...

23/257=0,0001011011101001...

24/257=0,0001011111101000...

25/257=0,0001100011100111...

26/257=0,0001100111100110...

27/257=0,0001101011100101...

28/257=0,0001101111100100...

29/257=0,0001110011100011...

30/257=0,0001110111100010...

31/257=0,0001111011100001...

32/257=0,0001111111100000...

33/257=0,0010000011011111...

34/257=0,0010000111011110...

35/257=0,0010001011011101...

36/257=0,0010001111011100...

37/257=0,0010010011011011...

38/257=0,0010010111011010...

39/257=0,0010011011011001...

40/257=0,0010011111011000...

41/257=0,0010100011010111...

42/257=0,0010100111010110...

43/257=0,0010101011010101...

44/257=0,0010101111010100...

45/257=0,0010110011010011...

46/257=0,0010110111010010...

47/257=0,0010111011010001...

48/257=0,0010111111010000...

49/257=0,0011000011001111...

50/257=0,0011000111001110...

51/257=0,0011001011001101...

52/257=0,0011001111001100...

53/257=0,0011010011001011...

54/257=0,0011010111001010...

55/257=0,0011011011001001...

56/257=0,0011011111001000...

57/257=0,0011100011000111...

58/257=0,0011100111000110...

59/257=0,0011101011000101...

60/257=0,0011101111000100...

61/257=0,0011110011000011...

62/257=0,0011110111000010...

63/257=0,0011111011000001...

64/257=0,0011111111000000...

65/257=0,0100000010111111...

66/257=0,0100000110111110...

67/257=0,0100001010111101...

68/257=0,0100001110111100...

69/257=0,0100010010111011...

70/257=0,0100010110111010...

71/257=0,0100011010111001...

72/257=0,0100011110111000...

73/257=0,0100100010110111...

74/257=0,0100100110110110...

75/257=0,0100101010110101...

76/257=0,0100101110110100...

77/257=0,0100110010110011...

78/257=0,0100110110110010...

79/257=0,0100111010110001...

80/257=0,0100111110110000...

81/257=0,0101000010101111...

82/257=0,0101000110101110...

83/257=0,0101001010101101...

84/257=0,0101001110101100...

85/257=0,0101010010101011...

86/257=0,0101010110101010...

87/257=0,0101011010101001...

88/257=0,0101011110101000...

89/257=0,0101100010100111...

90/257=0,0101100110100110...

91/257=0,0101101010100101...

92/257=0,0101101110100100...

93/257=0,0101110010100011...

94/257=0,0101110110100010...

95/257=0,0101111010100001...

96/257=0,0101111110100000...

97/257=0,0110000010011111...

98/257=0,0110000110011110...

99/257=0,0110001010011101...

100/257=0,0110001110011100...

101/257=0,0110010010011011...

102/257=0,0110010110011010...

103/257=0,0110011010011001...

104/257=0,0110011110011000...

105/257=0,0110100010010111...

106/257=0,0110100110010110...

107/257=0,0110101010010101...

108/257=0,0110101110010100...

109/257=0,0110110010010011...

110/257=0,0110110110010010...

111/257=0,0110111010010001...

112/257=0,0110111110010000...

113/257=0,0111000010001111...

114/257=0,0111000110001110...

115/257=0,0111001010001101...

116/257=0,0111001110001100...

117/257=0,0111010010001011...

118/257=0,0111010110001010...

119/257=0,0111011010001001...

120/257=0,0111011110001000...

121/257=0,0111100010000111...

122/257=0,0111100110000110...

123/257=0,0111101010000101...

124/257=0,0111101110000100...

125/257=0,0111110010000011...

126/257=0,0111110110000010...

127/257=0,0111111010000001...

128/257=0,0111111110000000...

129/257=0,1000000001111111...

130/257=0,1000000101111110...

131/257=0,1000001001111101...

132/257=0,1000001101111100...

133/257=0,1000010001111011...

134/257=0,1000010101111010...

135/257=0,1000011001111001...

136/257=0,1000011101111000...

137/257=0,1000100001110111...

138/257=0,1000100101110110...

139/257=0,1000101001110101...

140/257=0,1000101101110100...

141/257=0,1000110001110011...

142/257=0,1000110101110010...

143/257=0,1000111001110001...

144/257=0,1000111101110000...

145/257=0,1001000001101111...

146/257=0,1001000101101110...

147/257=0,1001001001101101...

148/257=0,1001001101101100...

149/257=0,1001010001101011...

150/257=0,1001010101101010...

151/257=0,1001011001101001...

152/257=0,1001011101101000...

153/257=0,1001100001100111...

154/257=0,1001100101100110...

155/257=0,1001101001100101...

156/257=0,1001101101100100...

157/257=0,1001110001100011...

158/257=0,1001110101100010...

159/257=0,1001111001100001...

160/257=0,1001111101100000...

161/257=0,1010000001011111...

162/257=0,1010000101011110...

163/257=0,1010001001011101...

164/257=0,1010001101011100...

165/257=0,1010010001011011...

166/257=0,1010010101011010...

167/257=0,1010011001011001...

168/257=0,1010011101011000...

169/257=0,1010100001010111...

170/257=0,1010100101010110...

171/257=0,1010101001010101...

172/257=0,1010101101010100...

173/257=0,1010110001010011...

174/257=0,1010110101010010...

175/257=0,1010111001010001...

176/257=0,1010111101010000...

177/257=0,1011000001001111...

178/257=0,1011000101001110...

179/257=0,1011001001001101...

180/257=0,1011001101001100...

181/257=0,1011010001001011...

182/257=0,1011010101001010...

183/257=0,1011011001001001...

184/257=0,1011011101001000...

185/257=0,1011100001000111...

186/257=0,1011100101000110...

187/257=0,1011101001000101...

188/257=0,1011101101000100...

189/257=0,1011110001000011...

190/257=0,1011110101000010...

191/257=0,1011111001000001...

192/257=0,1011111101000000...

193/257=0,1100000000111111...

194/257=0,1100000100111110...

195/257=0,1100001000111101...

196/257=0,1100001100111100...

197/257=0,1100010000111011...

198/257=0,1100010100111010...

199/257=0,1100011000111001...

200/257=0,1100011100111000...

201/257=0,1100100000110111...

202/257=0,1100100100110110...

203/257=0,1100101000110101...

204/257=0,1100101100110100...

205/257=0,1100110000110011...

206/257=0,1100110100110010...

207/257=0,1100111000110001...

208/257=0,1100111100110000...

209/257=0,1101000000101111...

210/257=0,1101000100101110...

211/257=0,1101001000101101...

212/257=0,1101001100101100...

213/257=0,1101010000101011...

214/257=0,1101010100101010...

215/257=0,1101011000101001...

216/257=0,1101011100101000...

217/257=0,1101100000100111...

218/257=0,1101100100100110...

219/257=0,1101101000100101...

220/257=0,1101101100100100...

221/257=0,1101110000100011...

222/257=0,1101110100100010...

223/257=0,1101111000100001...

224/257=0,1101111100100000...

225/257=0,1110000000011111...

226/257=0,1110000100011110...

227/257=0,1110001000011101...

228/257=0,1110001100011100...

229/257=0,1110010000011011...

230/257=0,1110010100011010...

231/257=0,1110011000011001...

232/257=0,1110011100011000...

233/257=0,1110100000010111...

234/257=0,1110100100010110...

235/257=0,1110101000010101...

236/257=0,1110101100010100...

237/257=0,1110110000010011...

238/257=0,1110110100010010...

239/257=0,1110111000010001...

240/257=0,1110111100010000...

241/257=0,1111000000001111...

242/257=0,1111000100001110...

243/257=0,1111001000001101...

244/257=0,1111001100001100...

245/257=0,1111010000001011...

246/257=0,1111010100001010...

247/257=0,1111011000001001...

248/257=0,1111011100001000...

249/257=0,1111100000000111...

250/257=0,1111100100000110...

251/257=0,1111101000000101...

252/257=0,1111101100000100...

253/257=0,1111110000000011...

254/257=0,1111110100000010...

255/257=0,1111111000000001...

256/257=0,1111111100000000...

On remarque que le produit du nombre de périodes (16) et de leurs longueurs (16) est égal à 256 et donc au premier -1.