[Wikipedia lists two articles on maximum length fo][1]r the continued fraction of $\sqrt n,$ Hickerson 1973 and Cohn 1977. There is a mess in the references, Cohn is not visible, just the link. Hmmm. This guy does a review of one of the (Russian) references, http://www.jpr2718.org/podstotalc.pdf I did my own modelling for two items, longest Gauss-Lagrange cycles of reduced forms, and largest number of reduced forms for a discriminant (positive but not a square). Reduced forms are $\langle a,b,c\rangle$ such that $\gcd(a,b,c) = 1, \; $ $d = b^2 - 4ac, \; \; $ $ac < 0, \; b > |a+c|.$ This equivalent version of reduction is [in Franz's book, Theorem 1.36, formula 1.34.][2] It is page numbered 37, pdf page 43. Details: the longest individual cycles were all prime discriminant, class number one. The largest form count was sometimes composite discriminant, class number not one. Hickerson and Cohn say that the period length for $\sqrt d$ is below $\sqrt d \log d.$ My own computations say this for both cycles of arbitrary redcued forms and total count of fomrs, where the latter has slightly larger implied constant (maybe over Cohn's $\frac{7}{2 \pi^2}.$ So the questions are, is $C_j \sqrt d \log d$ a provable upper bound for my two related problems? Best length of individual cycle: 5 1 1 -1 2 3.598813 0.5557389 17 1 3 -2 6 11.68164 0.5136266 41 1 5 -4 10 23.77846 0.4205486 73 1 7 -6 18 36.6577 0.4910291 193 1 13 -6 30 73.11163 0.4103314 241 1 15 -4 38 85.14695 0.4462873 337 1 17 -12 42 106.8425 0.3931019 409 1 19 -12 54 121.6199 0.4440064 601 1 23 -18 66 156.8635 0.420748 769 1 27 -10 70 184.274 0.3798691 1033 1 31 -18 78 223.061 0.3496802 1201 1 33 -28 106 245.7386 0.4313526 1609 1 39 -22 118 296.1642 0.3984276 1801 1 41 -30 130 318.1208 0.4086498 2161 1 45 -34 146 356.939 0.4090335 2521 1 49 -30 170 393.2619 0.4322819 3361 1 57 -28 178 470.7496 0.3781203 3529 1 59 -12 198 485.2689 0.4080212 4201 1 63 -58 210 540.7576 0.3883441 4561 1 67 -18 214 569.0039 0.3760958 5209 1 71 -42 238 617.6703 0.3853188 5569 1 73 -60 258 643.6448 0.4008422 6841 1 81 -70 290 730.3893 0.3970485 7561 1 85 -84 306 776.5653 0.3940428 8089 1 89 -42 330 809.2934 0.4077631 9241 1 95 -54 346 877.8031 0.3941658 12049 1 109 -42 378 1031.46 0.3664707 12289 1 109 -102 390 1043.869 0.3736102 12601 1 111 -70 394 1059.851 0.3717503 13729 1 117 -10 426 1116.317 0.3816119 15649 1 125 -6 454 1208.197 0.3757665 16921 1 129 -70 474 1266.507 0.3742578 18481 1 135 -64 502 1335.59 0.3758639 19009 1 137 -60 522 1358.418 0.3842705 20161 1 141 -70 530 1407.329 0.3765999 21121 1 145 -24 542 1447.206 0.3745147 21961 1 147 -88 566 1481.483 0.3820495 24049 1 155 -6 578 1564.397 0.3694714 26041 1 161 -30 590 1640.741 0.3595937 26161 1 161 -60 602 1645.26 0.3658996 28081 1 167 -48 622 1716.434 0.3623793 28729 1 169 -42 630 1739.992 0.3620706 31249 1 175 -156 674 1829.564 0.3683938 33049 1 181 -72 702 1891.701 0.3710947 33289 1 181 -132 714 1899.877 0.3758138 38329 1 195 -76 722 2066.233 0.3494282 40609 1 201 -52 750 2138.444 0.3507222 43201 1 207 -88 766 2218.496 0.345279 43801 1 209 -30 794 2236.735 0.3549817 47041 1 215 -204 842 2333.464 0.3608369 47881 1 217 -198 862 2358.079 0.3655518 48049 1 219 -22 878 2362.98 0.3715648 49009 1 221 -42 886 2390.848 0.3705798 51769 1 227 -60 914 2469.714 0.3700834 53881 1 231 -130 966 2528.87 0.3819888 59929 1 243 -220 974 2693.068 0.3616693 61681 1 247 -168 1002 2739.307 0.365786 65521 1 255 -124 1006 2838.747 0.3543817 66361 1 257 -78 1022 2860.168 0.3573217 67369 1 259 -72 1042 2885.721 0.3610882 69001 1 261 -220 1074 2926.753 0.3669596 70849 1 265 -156 1086 2972.721 0.3653219 80809 1 283 -180 1142 3212.198 0.3555198 87481 1 295 -114 1242 3365.64 0.3690234 101641 1 317 -288 1246 3675.646 0.338988 101929 1 319 -42 1270 3681.754 0.3449443 102001 1 319 -60 1298 3683.279 0.3524034 ========================================== Best total count of reduced forms: jagy@phobeusjunior:~$ d red red/(sqrt(d) log(d)) 5 2 0.555739 5 = 5 12 4 0.464686 12 = 2^2 * 3 17 6 0.513627 17 = 17 28 8 0.453711 28 = 2^2 * 7 41 10 0.420549 41 = 41 57 12 0.393129 57 = 3 * 19 73 18 0.491029 73 = 73 105 20 0.419385 105 = 3 * 5 * 7 145 28 0.467229 145 = 5 * 29 193 30 0.410331 193 = 193 217 32 0.403781 217 = 7 * 31 241 38 0.446287 241 = 241 265 40 0.440376 265 = 5 * 53 337 42 0.393102 337 = 337 385 44 0.376677 385 = 5 * 7 * 11 409 54 0.444006 409 = 409 481 56 0.413445 481 = 13 * 37 505 60 0.42894 505 = 5 * 101 601 66 0.420748 601 = 601 649 68 0.412209 649 = 11 * 59 721 72 0.407471 721 = 7 * 103 865 80 0.402217 865 = 5 * 173 889 84 0.414909 889 = 7 * 127 1009 90 0.409635 1009 = 1009 1081 92 0.400561 1081 = 23 * 47 1129 102 0.431871 1129 = 1129 1201 106 0.431353 1201 = 1201 1489 114 0.404377 1489 = 1489 1609 118 0.398428 1609 = 1609 1801 130 0.40865 1801 = 1801 1969 140 0.415943 1969 = 11 * 179 2161 146 0.409034 2161 = 2161 2521 170 0.432282 2521 = 2521 3241 180 0.391135 3241 = 7 * 463 3529 198 0.408021 3529 = 3529 3649 208 0.419803 3649 = 41 * 89 4201 210 0.388344 4201 = 4201 4321 216 0.392529 4321 = 29 * 149 4369 220 0.397072 4369 = 17 * 257 4729 230 0.395273 4729 = 4729 5209 238 0.385319 5209 = 5209 5401 240 0.379981 5401 = 11 * 491 5569 258 0.400842 5569 = 5569 6049 264 0.389817 6049 = 23 * 263 6169 272 0.396809 6169 = 31 * 199 6769 276 0.380341 6769 = 7 * 967 6841 290 0.397049 6841 = 6841 7561 306 0.394043 7561 = 7561 8089 330 0.407763 8089 = 8089 9241 346 0.394166 9241 = 9241 9529 352 0.393572 9529 = 13 * 733 10921 380 0.391059 10921 = 67 * 163 12289 390 0.37361 12289 = 12289 12601 394 0.37175 12601 = 12601 12961 404 0.374736 12961 = 13 * 997 13729 426 0.381612 13729 = 13729 14281 434 0.37962 14281 = 14281 14569 448 0.387165 14569 = 17 * 857 15409 472 0.394326 15409 = 19 * 811 15961 480 0.392582 15961 = 11 * 1451 17329 492 0.382933 17329 = 13 * 31 * 43 18001 516 0.392515 18001 = 47 * 383 19009 522 0.38427 19009 = 19009 20161 530 0.3766 20161 = 20161 20689 532 0.372195 20689 = 17 * 1217 21121 542 0.374515 21121 = 21121 21961 566 0.38205 21961 = 21961 23689 574 0.370245 23689 = 23689 23809 576 0.370412 23809 = 29 * 821 23881 584 0.374878 23881 = 11 * 13 * 167 25249 596 0.370028 25249 = 7 * 3607 26161 602 0.3659 26161 = 26161 27049 616 0.367007 27049 = 11 * 2459 28081 622 0.362379 28081 = 28081 28681 660 0.379691 28681 = 23 * 29 * 43 31201 680 0.372014 31201 = 41 * 761 33049 702 0.371095 33049 = 33049 33289 714 0.375814 33289 = 33289 37129 752 0.3709 37129 = 107 * 347 37801 756 0.368915 37801 = 103 * 367 40441 768 0.360025 40441 = 37 * 1093 40681 776 0.362499 40681 = 17 * 2393 43801 794 0.354982 43801 = 43801 43849 808 0.361006 43849 = 13 * 3373 44209 812 0.361037 44209 = 11 * 4019 44641 826 0.365148 44641 = 44641 45049 848 0.372856 45049 = 19 * 2371 46561 852 0.36735 46561 = 101 * 461 47881 862 0.365552 47881 = 47881 48049 878 0.371565 48049 = 48049 49009 886 0.37058 49009 = 49009 50521 912 0.374649 50521 = 19 * 2659 51769 914 0.370083 51769 = 51769 53881 966 0.381989 53881 = 53881 58969 984 0.368886 58969 = 109 * 541 61681 1002 0.365786 61681 = 61681 63361 1022 0.367213 63361 = 63361 65209 1040 0.367392 65209 = 61 * 1069 65641 1052 0.370186 65641 = 41 * 1601 69001 1074 0.36696 69001 = 69001 70849 1086 0.365322 70849 = 70849 74281 1100 0.359858 74281 = 59 * 1259 74881 1128 0.367273 74881 = 103 * 727 77401 1144 0.365291 77401 = 17 * 29 * 157 81481 1180 0.365564 81481 = 17 * 4793 84529 1200 0.363814 84529 = 137 * 617 85801 1212 0.364239 85801 = 239 * 359 86641 1252 0.374111 86641 = 23 * 3767 92569 1286 0.369611 92569 = 92569 95209 1296 0.366384 95209 = 19 * 5011 100321 1320 0.361886 100321 = 13 * 7717 d red red/(sqrt(d) log(d)) jagy@phobeusjunior:~$ [![enter image description here][3]][2] [1]: https://en.wikipedia.org/wiki/Periodic_continued_fraction#Length_of_the_repeating_block [2]: http://www.rzuser.uni-heidelberg.de/~hb3/publ/bf.pdf [3]: https://i.sstatic.net/3d6kp.jpg