Continued Fraction Expansions of Integral Square Roots

I have been fascinated by irrational numbers ever since I first saw Euclid's proof that the square root of two cannot be expressed as a rational fraction. Among all the irrational numbers, the square roots of the integers occupy a special place, because they can be expressed as continued fractions with a repeating period of finite length. In fact, the square root of any rational number can be expressed as a regular periodic continued fraction. These expansions are not only periodic; they are, except for the value marking the end of each period, always palindromic.

I discovered these facts for myself about 15 years ago, while playing around with an algorithm that allows one to construct the regular continued fraction expansion of n, where n is any natural number. Later I read up on the subject, and learned that Joseph Louis Lagrange had plowed this ground pretty thoroughly about 250 years ago, ca.1766. I have worked through the proofs of the assertions made above (about periods, and palindromes), but I don't have the time or the inclination to put that stuff on my web site right now. Maybe some day. Today, I'm just aiming to present some concrete data: the periodic expansions of the square roots of the first 400 positive integers, listed in tabular form below. (Click here for a printable table in portable document format. If your monitor is in the old 4:3 aspect ratio -- generally 1024 x 768 -- or even some weird contraption like 5:4 & 1280 x 960 -- and the table doesn't look good in your browser, you may want to visit this alternative page.)

In the following table "n" is an integer, "Continued Fraction" is the integer part of n (set off by a semicolon) followed by one or two repetitions of the periodic part of the fraction, and "Period" is the length of the periodic part of the canonical continued fraction for n. So, for example, the entry for n = 19 reads "4; 2, 1, 3, 1, 2, 8, ..." with a period of 6. Those entries represent the continued fraction

Canonical Continued Fraction for Sqrt(19)

where 4 is the integer part of √19, and the sequence of partial denominators {2, 1, 3, 1, 2, 8} repeats ad infinitum.




n Continued Fraction Period n Continued Fraction Period n Continued Fraction Period n Continued Fraction Period n Continued Fraction Period
1 Perfect Square 0 2 1; 2, 2, ... 1 3 1; 1, 2, 1, 2, ... 2 4 Perfect Square 0 5 2; 4, 4, ... 1
6 2; 2, 4, 2, 4, ... 2 7 2; 1, 1, 1, 4, ... 4 8 2; 1, 4, 1, 4, ... 2 9 Perfect Square 0 10 3; 6, 6, ... 1
11 3; 3, 6, 3, 6, ... 2 12 3; 2, 6, 2, 6, ... 2 13 3; 1, 1, 1, 1, 6, ... 5 14 3; 1, 2, 1, 6, ... 4 15 3; 1, 6, 1, 6, ... 2
16 Perfect Square 0 17 4; 8, 8, ... 1 18 4; 4, 8, 4, 8, ... 2 19 4; 2, 1, 3, 1, 2, 8, ... 6 20 4; 2, 8, 2, 8, ... 2
21 4; 1, 1, 2, 1, 1, 8, ... 6 22 4; 1, 2, 4, 2, 1, 8, ... 6 23 4; 1, 3, 1, 8, ... 4 24 4; 1, 8, 1, 8, ... 2 25 Perfect Square 0
26 5; 10, 10, ... 1 27 5; 5, 10, 5, 10, ... 2 28 5; 3, 2, 3, 10, ... 4 29 5; 2, 1, 1, 2, 10, ... 5 30 5; 2, 10, 2, 10, ... 2
31 5; 1, 1, 3, 5, 3, 1, 1, 10, ... 8 32 5; 1, 1, 1, 10, ... 4 33 5; 1, 2, 1, 10, ... 4 34 5; 1, 4, 1, 10, ... 4 35 5; 1, 10, 1, 10, ... 2
36 Perfect Square 0 37 6; 12, 12, ... 1 38 6; 6, 12, 6, 12, ... 2 39 6; 4, 12, 4, 12, ... 2 40 6; 3, 12, 3, 12, ... 2
41 6; 2, 2, 12, 2, 2, 12, ... 3 42 6; 2, 12, 2, 12, ... 2 43 6; 1, 1, 3, 1, 5, 1, 3, 1, 1, 12, ... 10 44 6; 1, 1, 1, 2, 1, 1, 1, 12, ... 8 45 6; 1, 2, 2, 2, 1, 12, ... 6
46 6; 1,3,1,1,2,6,2,1,1,3,1,12,... 12 47 6; 1, 5, 1, 12, ... 4 48 6; 1, 12, 1, 12, ... 2 49 Perfect Square 0 50 7; 14, 14, ... 1
51 7; 7, 14, 7, 14, ... 2 52 7; 4, 1, 2, 1, 4, 14, ... 6 53 7; 3, 1, 1, 3, 14, ... 5 54 7; 2, 1, 6, 1, 2, 14, ... 6 55 7; 2, 2, 2, 14, ... 4
56 7; 2, 14, 2, 14, ... 2 57 7; 1, 1, 4, 1, 1, 14, ... 6 58 7; 1, 1, 1, 1, 1, 1, 14, ... 7 59 7; 1, 2, 7, 2, 1, 14, ... 6 60 7; 1, 2, 1, 14, ... 4
61 7; 1,4,3,1,2,2,1,3,4,1,14,... 11 62 7; 1, 6, 1, 14, ... 4 63 7; 1, 14, 1, 14, ... 2 64 Perfect Square 0 65 8; 16, 16, ... 1
66 8; 8, 16, 8, 16, ... 2 67 8; 5, 2, 1, 1, 7, 1, 1, 2, 5, 16, ... 10 68 8; 4, 16, 4, 16, ... 2 69 8; 3, 3, 1, 4, 1, 3, 3, 16, ... 8 70 8; 2, 1, 2, 1, 2, 16, ... 6
71 8; 2, 2, 1, 7, 1, 2, 2, 16, ... 8 72 8; 2, 16, 2, 16, ... 2 73 8; 1, 1, 5, 5, 1, 1, 16, ... 7 74 8; 1, 1, 1, 1, 16, ... 5 75 8; 1, 1, 1, 16, ... 4
76 8; 1,2,1,1,5,4,5,1,1,2,1,16,... 12 77 8; 1, 3, 2, 3, 1, 16, ... 6 78 8; 1, 4, 1, 16, ... 4 79 8; 1, 7, 1, 16, ... 4 80 8; 1, 16, 1, 16, ... 2
81 Perfect Square 0 82 9; 18, 18, ... 1 83 9; 9, 18, 9, 18, ... 2 84 9; 6, 18, 6, 18, ... 2 85 9; 4, 1, 1, 4, 18, ... 5
86 9; 3, 1, 1, 1, 8, 1, 1, 1, 3, 18, ... 10 87 9; 3, 18, 3, 18, ... 2 88 9; 2, 1, 1, 1, 2, 18, ... 6 89 9; 2, 3, 3, 2, 18, ... 5 90 9; 2, 18, 2, 18, ... 2
91 9; 1, 1, 5, 1, 5, 1, 1, 18, ... 8 92 9; 1, 1, 2, 4, 2, 1, 1, 18, ... 8 93 9; 1, 1, 1, 4, 6, 4, 1, 1, 1, 18, ... 10 94 9; 1,2,3,1,1,5,1,8,1,5,1,1,3,2,1,18,... 16 95 9; 1, 2, 1, 18, ... 4
96 9; 1, 3, 1, 18, ... 4 97 9; 1,5,1,1,1,1,1,1,5,1,18,... 11 98 9; 1, 8, 1, 18, ... 4 99 9; 1, 18, 1, 18, ... 2 100 Perfect Square 0
101 10; 20, 20, ... 1 102 10; 10, 20, 10, 20, ... 2 103 10; 6,1,2,1,1,9,1,1,2,1,6,20,... 12 104 10; 5, 20, 5, 20, ... 2 105 10; 4, 20, 4, 20, ... 2
106 10; 3, 2, 1, 1, 1, 1, 2, 3, 20, ... 9 107 10; 2, 1, 9, 1, 2, 20, ... 6 108 10; 2, 1, 1, 4, 1, 1, 2, 20, ... 8 109 10; 2,3,1,2,4,1,6,6,1,4,2,1,3,2,20,... 15 110 10; 2, 20, 2, 20, ... 2
111 10; 1, 1, 6, 1, 1, 20, ... 6 112 10; 1, 1, 2, 1, 1, 20, ... 6 113 10; 1, 1, 1, 2, 2, 1, 1, 1, 20, ... 9 114 10; 1, 2, 10, 2, 1, 20, ... 6 115 10; 1, 2, 1, 1, 1, 1, 1, 2, 1, 20, ... 10
116 10; 1, 3, 2, 1, 4, 1, 2, 3, 1, 20, ... 10 117 10; 1, 4, 2, 4, 1, 20, ... 6 118 10; 1, 6, 3, 2, 10, 2, 3, 6, 1, 20, ... 10 119 10; 1, 9, 1, 20, ... 4 120 10; 1, 20, 1, 20, ... 2
121 Perfect Square 0 122 11; 22, 22, ... 1 123 11; 11, 22, 11, 22, ... 2 124 11; 7,2,1,1,1,3,1,4,1,3,1,1,1,2,7,22,... 16 125 11; 5, 1, 1, 5, 22, ... 5
126 11; 4, 2, 4, 22, ... 4 127 11; 3,1,2,2,7,11,7,2,2,1,3,22,... 12 128 11; 3, 5, 3, 22, ... 4 129 11; 2, 1, 3, 1, 6, 1, 3, 1, 2, 22, ... 10 130 11; 2, 2, 22, ... 3
131 11; 2, 4, 11, 4, 2, 22, ... 6 132 11; 2, 22, 2, 22, ... 2 133 11; 1,1,7,5,1,1,1,2,1,1,1,5,7,1,1,22,... 16 134 11; 1,1,2,1,3,1,10,1,3,1,2,1,1,22,... 14 135 11; 1, 1, 1, 1, 1, 1, 1, 22, ... 8
136 11; 1, 1, 1, 22, ... 4 137 11; 1, 2, 2, 1, 1, 2, 2, 1, 22, ... 9 138 11; 1, 2, 1, 22, ... 4 139 11; 1,3,1,3,7,1,1,2,11,2,1,1,7,3,1,3,1,22,... 18 140 11; 1, 4, 1, 22, ... 4
141 11; 1, 6, 1, 22, ... 4 142 11; 1, 10, 1, 22, ... 4 143 11; 1, 22, 1, 22, ... 2 144 Perfect Square 0 145 12; 24, 24, ... 1
146 12; 12, 24, 12, 24, ... 2 147 12; 8, 24, 8, 24, ... 2 148 12; 6, 24, 6, 24, ... 2 149 12; 4, 1, 5, 3, 3, 5, 1, 4, 24, ... 9 150 12; 4, 24, 4, 24, ... 2
151 12; 3,2,7,1,3,4,1,1,1,11,1,1,1,4,3,1,7,2,3,24,... 20 152 12; 3, 24, 3, 24, ... 2 153 12; 2, 1, 2, 2, 2, 1, 2, 24, ... 8 154 12; 2, 2, 3, 1, 2, 1, 3, 2, 2, 24, ... 10 155 12; 2, 4, 2, 24, ... 4
156 12; 2, 24, 2, 24, ... 2 157 12; 1,1,7,1,5,2,1,1,1,1,2,5,1,7,1,1,24,... 17 158 12; 1, 1, 3, 12, 3, 1, 1, 24, ... 8 159 12; 1, 1, 1, 1, 3, 1, 1, 1, 1, 24, ... 10 160 12; 1, 1, 1, 5, 1, 1, 1, 24, ... 8
161 12; 1, 2, 4, 1, 2, 1, 4, 2, 1, 24, ... 10 162 12; 1, 2, 1, 2, 12, 2, 1, 2, 1, 24, ... 10 163 12; 1,3,3,2,1,1,7,1,11,1,7,1,1,2,3,3,1,24,... 18 164 12; 1, 4, 6, 4, 1, 24, ... 6 165 12; 1, 5, 2, 5, 1, 24, ... 6
166 12; 1,7,1,1,1,2,4,1,3,2,12,2,3,1,4,2,1,1,1,7,1,24,... 22 167 12; 1, 11, 1, 24, ... 4 168 12; 1, 24, 1, 24, ... 2 169 Perfect Square 0 170 13; 26, 26, ... 1
171 13; 13, 26, 13, 26, ... 2 172 13; 8,1,2,2,1,1,3,6,3,1,1,2,2,1,8,26,... 16 173 13; 6, 1, 1, 6, 26, ... 5 174 13; 5, 4, 5, 26, ... 4 175 13; 4, 2, 1, 2, 4, 26, ... 6
176 13; 3, 1, 3, 26, ... 4 177 13; 3, 3, 2, 8, 2, 3, 3, 26, ... 8 178 13; 2, 1, 12, 1, 2, 26, ... 6 179 13; 2,1,1,1,3,5,13,5,3,1,1,1,2,26,... 14 180 13; 2, 2, 2, 26, ... 4
181 13; 2,4,1,8,6,1,1,1,1,2,2,1,1,1,1,6,8,1,4,2,26,... 21 182 13; 2, 26, 2, 26, ... 2 183 13; 1, 1, 8, 1, 1, 26, ... 6 184 13; 1,1,3,2,1,2,1,2,3,1,1,26,... 12 185 13; 1, 1, 1, 1, 26, ... 5
186 13; 1, 1, 1, 3, 4, 3, 1, 1, 1, 26, ... 10 187 13; 1, 2, 13, 2, 1, 26, ... 6 188 13; 1, 2, 2, 6, 2, 2, 1, 26, ... 8 189 13; 1, 2, 1, 26, ... 4 190 13; 1,3,1,1,1,2,2,2,1,1,1,3,1,26,... 14
191 13; 1,4,1,1,3,2,2,13,2,2,3,1,1,4,1,26,... 16 192 13; 1, 5, 1, 26, ... 4 193 13; 1,8,3,2,1,3,3,1,2,3,8,1,26,... 13 194 13; 1, 12, 1, 26, ... 4 195 13; 1, 26, 1, 26, ... 2
196 Perfect Square 0 197 14; 28, 28, ... 1 198 14; 14, 28, 14, 28, ... 2 199 14; 9,2,1,2,2,5,4,1,1,13,1,1,4,5,2,2,1,2,9,28,... 20 200 14; 7, 28, 7, 28, ... 2
201 14; 5,1,1,1,2,1,8,1,2,1,1,1,5,28,... 14 202 14; 4, 1, 2, 2, 1, 4, 28, ... 7 203 14; 4, 28, 4, 28, ... 2 204 14; 3, 1, 1, 6, 1, 1, 3, 28, ... 8 205 14; 3, 6, 2, 4, 1, 6, 3, 28, ... 8
206 14; 2, 1, 5, 14, 5, 1, 2, 28, ... 8 207 14; 2, 1, 1, 2, 1, 1, 2, 28, ... 8 208 14; 2, 2, 1, 2, 2, 28, ... 6 209 14; 2, 5, 3, 2, 3, 5, 2, 28, ... 8 210 14; 2, 28, 2, 28, ... 2
211 14; 1,1,9,5,1,2,2,1,1,4,3,1,13,1,3,4,1,1,2,2,1,5,9,1,1,28,... 26 212 14; 1,1,3,1,1,1,6,1,1,1,3,1,1,28,... 14 213 14; 1,1,2,6,1,8,1,6,2,1,1,28,... 12 214 14; 1,1,1,2,3,1,4,9,1,1,5,3,14,3,5,1,1,9,4,1,3,2,1,1,1,28,... 26 215 14; 1, 1, 1, 28, ... 4
216 14; 1, 2, 3, 2, 1, 28, ... 6 217 14; 1,2,1,2,1,1,9,4,9,1,1,2,1,2,1,28,... 16 218 14; 1, 3, 3, 1, 28, ... 5 219 14; 1, 3, 1, 28, ... 4 220 14; 1, 4, 1, 28, ... 4
221 14; 1, 6, 2, 6, 1, 28, ... 6 222 14; 1, 8, 1, 28, ... 4 223 14; 1, 13, 1, 28, ... 4 224 14; 1, 28, 1, 28, ... 2 225 Perfect Square 0
226 15; 30, 30, ... 1 227 15; 15, 30, 15, 30, ... 2 228 15; 10, 30, 10, 30, ... 2 229 15; 7, 1, 1, 7, 30, ... 5 230 15; 6, 30, 6, 30, ... 4
231 15; 5, 30, 5, 30, ... 2 232 15; 4, 3, 7, 3, 4, 30, ... 6 233 15; 3, 1, 3, 1, 1, 1, 1, 3, 1, 3, 30,... 11 234 15; 3, 2, 1, 2, 1, 2, 3, 30, ... 8 235 15; 3, 30, 3, 30, ... 2
236 15; 2,1,3,5,1,6,1,5,3,1,2,30,... 12 237 15; 2, 1, 1, 7, 10, 7, 1, 1, 2, 30, ... 10 238 15; 2, 2, 1, 14, 1, 2, 2, 30, ... 8 239 15; 2,5,1,2,4,15,4,2,1,5,2,30,... 12 240 15; 2, 30, 2, 30, ... 2
241 15; 1,1,9,1,5,3,3,1,1,3,3,5,1,9,1,1,30,... 17 242 15; 1, 1, 3, 1, 14, 1, 3, 1, 1, 30, ... 10 243 15; 1, 1, 2, 3, 15, 3, 2, 1, 1, 30, ... 10 244 15; 1,1,1,1,1,2,1,5,1,1,9,1,6,1,9,1,1,5,1,2,1,1,1,1,1,30,... 26 245 15; 1, 1, 1, 7, 6, 7, 1, 1, 1, 30, ... 10
246 15; 1, 1, 2, 3, 15, 3, 2, 1, 1, 30, ... 10 247 15; 1,2,1,1,9,1,9,1,1,2,1,30,... 12 248 15; 1, 2, 1, 30, ... 4 249 15; 1,3,1,1,5,1,3,10,3,1,5,1,1,3,1,30,... 16 250 15; 1, 4, 3, 3, 4, 1, 30, ... 7
251 15; 1,5,2,1,2,2,15,2,2,1,2,5,1,30,... 14 252 15; 1, 6, 1, 30, ... 4 253 See Note 1a, infra 22 254 15; 1, 14, 1, 30, ... 4 255 15; 1, 30, 1, 30, ... 2
256 Perfect Square 0 257 16; 32, 32, ... 1 258 16; 16, 32, 16, 32, ... 2 259 16; 10, 1, 2, 3, 4, 3, 2, 1, 10, 32, ... 10 260 16; 8, 32, 8, 32, ... 2
261 16; 6,2,3,7,1,3,1,2,1,3,1,7,3,2,6,32,... 16 262 16; 5,2,1,2,1,10,16,101,2,1,2,5,32,... 14 263 16; 4,1,1,1,1,15,1,1,1,1,4,32,... 12 264 16; 4, 32, 4, 32, ... 2 265 16; 3, 1, 1, 2, 2, 1, 1, 3, 32, ... 9
266 16; 3, 4, 3, 32, ... 4 267 16; 2, 1, 15, 1, 2, 32, ... 6 268 See Note 2a, infra 20 269 16; 2, 2, 32, ... 3 270 16; 2, 3, 6, 3, 2, 32, ... 6
271 16; 2,6,10,1,4,1,1,2,1,2,1,15,1,2,1,2,1,1,4,1,10,6,2,32,... 24 272 16; 2, 32, 2, 32, ... 2 273 16; 1, 1, 10, 1, 1, 32, ... 6 274 16; 1, 1, 4, 4, 1, 1, 32, ... 7 275 16; 1, 1, 2, 1, 1, 32, ... 6
276 16; 1,1,1,1,2,2,2,1,1,1,1,32, ... 12 277 See Note 2b, infra 21 278 16; 1, 2, 16, 2, 1, 32, ... 6 279 16; 1, 2, 2, 1, 2, 2, 1, 32, ... 8 280 16; 1, 2, 1, 2, 1, 32, ... 6
281 16; 1,3,4,1,1,6,6,1,1,4,3,1,32,... 13 282 16; 1, 3, 1, 4, 1, 3, 1, 32, ... 8 283 16; 1, 2, 1, 2, 1, 32, ... 6 284 16; 1,5,1,3,2,1,4,8,4,1,2,3,1,5,1,32,... 16 285 16; 1, 7, 2, 7, 1, 32, ... 6
286 16; 1, 10, 3, 3, 2, 3, 3, 10, 1, 32, ... 10 287 16; 1, 15, 1, 32, ... 4 288 16; 1, 32, 1, 32, ... 2 289 Perfect Square 0 290 17; 34, 34, ... 1
291 17; 17, 34, 17, 34, ... 2 292 17; 11, 2, 1, 3, 8, 3, 1, 2, 11, 34, ... 10 293 17; 8, 1, 1, 8, 34, ... 5 294 17; 6, 1, 4, 1, 6, 34, ... 6 295 17; 5,1,2,3,2,6,2,3,2,1,5,34,... 12
296 17; 4, 1, 7, 1, 34, ... 6 297 17; 4, 3, 1, 1, 2, 1, 1, 3, 4, 34, ... 10 298 17; 3,1,4,5,1,1,5,4,1,3,34,... 11 299 17; 3, 2, 3, 34, ... 4 300 17; 3, 8, 3, 34, ... 4
301 17; 2,1,6,3,1,2,2,1,1,8,11,2,4,2,11,8,1,1,2,2,1,3,6,1,2,34,... 26 302 17; 2,1,1,1,4,2,1,16,1,2,4,1,1,1,2,34,... 16 303 17; 2, 2, 5, 2, 2, 34, ... 6 304 17; 2,3,2,1,1,1,1,1,2,3,2,34,... 12 305 17; 2, 6, 2, 34, ... 4
306 17; 2, 34, 2, 34, ... 2 307 17; 1,1,11,5,1,3,17,3,1,5,11,1,1,34,... 14 308 17; 1, 1, 4, 1, 1, 34, ... 6 309 17; 1,1,2,1,2,4,1,1,1,8,6,1,10,1,6,8,1,1,1,4,2,1,2,1,1,34,... 26 310 See note 3a, infra 16
311 17; 1,1,1,2,1,6,3,17,3,6,1,2,1,1,1,34,... 16 312 17; 1, 1, 1, 34, ... 4 313 17; 1,2,4,11,1,1,3,2,2,3,1,1,11,4,2,1,34,... 17 314 17; 1, 2, 1, 1, 2, 1, 34, ... 7 315 17; 1, 2, 1, 34, ... 4
316 17; 1, 3, 2, 8, 2, 3, 1, 34, ... 8 317 17; 1,4,8,1,2,2,1,8,4,1,34,... 11 318 17; 1, 4, 1, 34, ... 4 319 17; 1,6,5,1,4,3,1,3,4,1,5,6,1,34,... 14 320 17; 1, 7, 1, 34, ... 4
321 17; 1, 10, 1, 34, ... 4 322 17; 1, 16, 1, 34, ... 4 323 17; 1, 34, 1, 34, ... 2 324 Perfect Square 0 325 18; 36, 36, ... 1
326 18; 18, 36, 18, 16, ... 2 327 18; 12, 36, 12, 36, ... 2 328 18; 9, 36, 9, 36, ... 2 329 18; 7, 1, 1, 7, 30, ... 5 330 18; 6, 30, 6, 30, ... 4
331 See Note 4a, infra 34 332 18; 4, 1, 1, 8, 1, 1, 4, 36, ... 8 333 18; 4, 36, 4, 36,... 2 334 See Note 4b, infra 28 335 18; 3, 3, 3, 36, ... 4
336 18; 3, 36, 3, 36,... 2 337 See Note 4c, infra 19 338 18; 2, 1, 1, 2, 36, ... 5 339 18; 2, 2, 2, 1, 17, 1, 2, 2, 2, 36, ... 10 340 18; 2,3,1,1,1,1,8,1,1,1,1,3,2,36,... 14
341 18; 2,6,1,8,2,1,2,1,2,8,1,6,2,36,... 14 342 18; 2, 36, 2, 36, ... 2 343 18; 1,1,11,1,5,3,1,17,1,3,5,1,11,1,1,36, ... 16 344 18; 1, 1, 4, 1, 3, 1, 4, 1, 1, 36, ... 10 345 18; 1, 1, 2, 1, 6, 1, 2, 1, 1, 36, ... 10
346 18; 1, 1, 1, 1, 36, ... 5 347 18; 1,1,1,2,4,1,17,1,4,2,1,1,1,36,... 14 348 18; 1, 1, 1, 8, 1, 1, 1, 36, ... 8 349 18; 1, 2, 7, 7, 2, 1, 36, ... 7 350 18; 1,2,2,2,1,36, ... 6
351 18; 1,2,1,3,2,2,2,3,1,2,1,36,... 12 352 18; 1, 3, 5, 9, 5, 3, 1, 36, ... 8 353 18; 1,3,1,2,1,1,1,1,1,1,2,1,3,1,36,... 15 354 18; 1, 4, 2, 2, 18, 2, 2, 4, 1, 36, ... 10 355 18; 1,5,3,3,1,6,2,3,3,5,1,36, ... 12
356 18; 1,6,1,1,2,1,8,1,2,1,1,6,1,36,... 14 357 18; 1, 8, 2, 8, 1, 36, ... 6 358 See Note 4d, infra 26 359 18; 1, 17, 1, 36, ... 4 360 18; 1, 36, 1, 36, ... 2
361 Perfect Square 0 362 19; 38, 38, ... 14 363 19; 19, 38, 19, 38, ... 2 364 19; 12,1,2,3,1,8,1,3,2,1,12,38,... 12 365 19; 9, 1, 1, 9, 38, ... 5
366 19; 7,1,1,1,2,12,2,1,1,1,7,38, ... 12 367 See Note 5a, infra 20 368 19; 5, 2, 5, 38, ... 5 369 19; 4,1,3,2,7,4,7,2,3,1,4,38, ... 12 370 19; 4, 4, 38, ... 3
371 19; 3, 1, 4, 1, 3, 38, ... 6 372 19; 3, 2, 12, 2, 3, 38, ... 6 373 19; 3, 5, 5, 3, 38, ... 5 374 19; 2, 1, 18, 1, 2, 38, ... 6 375 19; 2, 1, 2, 1, 5, 1, 2, 1, 2, 38, ... 10
376 19; 2,1,1,3,1,2,2,4,2,2,1,3,1,1,2,38, ... 16 377 19; 2, 2, 2, 2, 38, ... 5 378 19; 2, 3, 1, 4, 1, 3, 2, 38, ... 8 379 See Note 5b, infra 30 380 19; 2, 38, ... 2
381 19; 1, 1, 12, 1, 1, 38, ... 8 382 See Note 5c, infra 24 383 19; 1, 1, 3, 19, 3, 1, 1, 38, ... 8 384 19; 1, 1, 2, 9, 2, 1, 1, 38, ... 8 385 See Note 5d, infra 16
386 19; 1,1,1,4,1,18,1,4,1,1,38,... 12 387 19; 1, 2, 19, 2, 1, 38, ... 6 388 19; 1,2,3,4,12,1,8,1,12,4,3,2,1,38,... 14 389 19; 1, 2, 1, 1, 1, 1, 2, 1, 38, ... 9 390 19; 1, 2, 1, 38, ... 4
391 19; 1,3,2,2,1,1,2,19,2,1,1,2,2,3,1,38,... 16 392 19; 1, 3, 1, 38, ... 4 393 See Note 5e, infra 20 394 19; 1,5,1,1,1,3,1,3,5,2,2,5,3,1,3,1,1,1,5,1,38,... 21 395 19; 1, 6, 1, 38, ... 4
396 19; 1, 8, 1, 38, ... 4 397 See Note 5f, infra 21 398 19; 1, 18, 1, 38,... 4 399 19; 1, 38, 1, 38, ... 2 400 Perfect Square 0
   Notes ...
   1a: 253 15; 1,9,1,1,1,2,1,7,4,2,2,2,4,7,1,2,1,1,1,9,1,30,...                             5a: 367 19; 6,2,1,3,1,1,2,1,12,19,12,1,2,1,1,3,1,2,6,38,...
   2a: 268 16; 2,1,2,3,3,1,3,1,10,8,10,1,3,1,3,3,2,1,2,32,...                               5b: 379 19; 2,7,3,2,2,6,12,1,4,1,1,1,3,4,19,4,3,1,1,1,4,1,12,6,2,2,3,7,2,38,...
   2b: 277 16; 1,1,1,4,10,1,7,2,2,3,3,2,2,7,1,10,4,1,1,1,32,...                             5c: 382 19; 1,1,5,12,1,5,1,1,2,3,1,18,1,3,2,1,1,5,1,12,5,1,1,38,...
   3a: 310 17; 1,1,1,1,5,3,1,2,1,3,5,1,1,1,1,34,...                                         5d: 385 19; 1,1,1,1,1,3,1,2,1,3,1,1,1,1,1,38,...
   4a: 331 18; 5,5,1,6,2,3,1,1,2,1,2,1,11,2,1,1,17,1,1,2,11,1,2,1,2,1,1,3,2,6,1,5,5,36,...  5e: 393 19; 1,4,1,2,4,1,1,1,1,12,1,1,1,1,4,2,1,4,1,38,...
   4b: 334 18; 3,1,1,1,2,5,1,2,2,11,1,3,7,18,7,3,1,11,2,2,1,5,2,1,1,1,3,36,...              5f: 397 19; 1,12,3,4,9,1,2,1,2,1,1,2,1,2,1,9,4,3,12,1,38,...
   4c: 337 18; 2,1,3,1,11,2,4,1,3,3,1,4,2,11,1,3,1,2,36,...
   4d: 358 18; 1,11,1,1,1,3,1,1,4,1,5,2,18,2,5,1,4,1,1,3,1,1,1,11,1,36,...