3
$\begingroup$

Wikipedia lists two articles on maximum length for 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. 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 reduced forms and total count of forms, 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: let d be the discriminant, a,b,c the coefficients of the form considered, L the length of the cycle, cohn = sqrt(d) log(d), and ratio = L/cohn

     d           a           b           c              L    cohn       ratio
     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
     d           a           b           c              L    cohn       ratio

==========================================

Best total count of reduced forms:

file named many_reduced_binary.txt 
        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
   105169      1328       0.354137         105169 = 251 * 419
   105361      1330       0.354291         105361 = 105361
   109729      1360       0.353756         109729 = 197 * 557
   111721      1378       0.354679         111721 = 111721
   114409      1396       0.354341         114409 = 191 * 599
   117049      1496       0.374683         117049 = 67 * 1747
   127009      1516       0.361968         127009 = 107 * 1187
   130729      1558       0.365766         130729 = 130729
   138121      1584       0.360101         138121 = 37 * 3733
   144169      1610        0.35696         144169 = 144169
   153169      1672       0.357826         153169 = 89 * 1721
   155401      1676       0.355666         155401 = 19 * 8179
   157249      1680       0.354064         157249 = 67 * 2347
   157921      1736       0.364956         157921 = 79 * 1999
   173209      1814       0.361346         173209 = 173209
   180049      1852       0.360682         180049 = 401 * 449
   185641      1902       0.363879         185641 = 185641
   197521      1910       0.352448         197521 = 197521
   204649      1916       0.346336         204649 = 19 * 10771
   204961      1948       0.351809         204961 = 97 * 2113
   210961      1978       0.351281         210961 = 210961
   220081      1992       0.345168         220081 = 29 * 7589
   223609      2104       0.361221         223609 = 311 * 719
   242449      2114       0.346277         242449 = 242449
   251329      2144       0.343934         251329 = 13 * 19333
   259009      2170       0.342077         259009 = 259009
   261361      2200       0.344992         261361 = 31 * 8431
   266809      2216       0.343367         266809 = 367 * 727
   268129      2272       0.351037         268129 = 73 * 3673
   279889      2316       0.349039         279889 = 19 * 14731
   287401      2396       0.355594         287401 = 31 * 73 * 127
   295201      2398       0.350411         295201 = 295201
   306289      2412       0.345008         306289 = 17 * 43 * 419
   314161      2446       0.344768         314161 = 314161
   315121      2452       0.345004         315121 = 181 * 1741
   316369      2476       0.347584         316369 = 19 * 16651
   320401      2534       0.353128         320401 = 320401
   328441      2600       0.357165         328441 = 53 * 6197
   349801      2602       0.344645         349801 = 349801
   357361      2656       0.347474         357361 = 191 * 1871
   360721      2664        0.34664         360721 = 137 * 2633
   367249      2692       0.346671         367249 = 181 * 2029
   373129      2700       0.344523         373129 = 23 * 16223
   383881      2716       0.340922         383881 = 103 * 3727
   389041      2730       0.340046         389041 = 389041
   393121      2782       0.344441         393121 = 393121
   394969      2866       0.353881         394969 = 394969
   415801      2878       0.344971         415801 = 415801
   416329      2936       0.351665         416329 = 263 * 1583
   425569      2952        0.34913         425569 = 23 * 18503
   457081      2996       0.340027         457081 = 83 * 5507
   463849      2998       0.337382         463849 = 463849
   465481      3008       0.337823         465481 = 19 * 24499
   469249      3032       0.338939         469249 = 11 * 29 * 1471
   469561      3038       0.339479         469561 = 469561
   483289      3154       0.346635         483289 = 483289
   503281      3212       0.344859         503281 = 463 * 1087
   511681      3244       0.344989         511681 = 23 * 22247
   515761      3330       0.352519         515761 = 515761
   528889      3336       0.348078         528889 = 167 * 3167
   540409      3348       0.345023         540409 = 439 * 1231
   552841      3386         0.3444         552841 = 552841
   563161      3432       0.345382         563161 = 233 * 2417
   579049      3448       0.341481         579049 = 31 * 18679
   600601      3542       0.343493         600601 = 600601
   601129      3560       0.345064         601129 = 509 * 1181
   605641      3570       0.344549         605641 = 605641
   628489      3668       0.346549         628489 = 41 * 15329
   663961      3706       0.339262         663961 = 663961
   677161      3728       0.337438         677161 = 17 * 61 * 653
   685441      3812       0.342641         685441 = 31 * 22111
   691681      3846       0.343902         691681 = 691681
   702529      3866       0.342615         702529 = 702529
   722929      3920       0.341737         722929 = 79 * 9151
   723529      3950       0.344188         723529 = 723529
   766921      4004       0.337423         766921 = 17 * 197 * 229
   772489      4024       0.337704         772489 = 31 * 24919
   786241      4134       0.343441         786241 = 786241
   799681      4136       0.340283         799681 = 37 * 21613
   819289      4174       0.338671         819289 = 819289
   825841      4256       0.343751         825841 = 379 * 2179
   878641      4398       0.342822         878641 = 878641
   914281      4416       0.336472         914281 = 337 * 2713
   924169      4444       0.336525         924169 = 881 * 1049
   928489      4528       0.341971         928489 = 17 * 54617
   969049      4590       0.338269         969049 = 969049
   979969      4698       0.344014         979969 = 979969
  1007161      4706       0.339243        1007161 = 1007161
  1057081      4792       0.336012        1057081 = 157 * 6733
  1083289      4966       0.343368        1083289 = 1083289
  1162561      4988        0.33124        1162561 = 103 * 11287
  1171921      5070       0.335145        1171921 = 1171921
  1195489      5118        0.33449        1195489 = 1195489
  1201561      5300       0.345384        1201561 = 541 * 2221
  1264561      5322       0.336838        1264561 = 1264561
  1267249      5424       0.342878        1267249 = 31 * 40879
  1354369      5468       0.332783        1354369 = 73 * 18553
  1354609      5480       0.333479        1354609 = 71 * 19079
  1368481      5532       0.334692        1368481 = 29 * 47189
  1379569      5576       0.335803        1379569 = 43 * 32083
  1391329      5624       0.337057        1391329 = 83 * 16763
  1433041      5826       0.343328        1433041 = 1433041
  1466089      5856       0.340636        1466089 = 23 * 63743
  1538161      5864       0.331892        1538161 = 1063 * 1447
  1552849      5900       0.332125        1552849 = 229 * 6781
  1577689      5910       0.329692        1577689 = 1577689
  1609609      5920         0.3265        1609609 = 23 * 47 * 1489
  1627921      6120       0.335362        1627921 = 571 * 2851
  1642369      6132       0.334331        1642369 = 433 * 3793
  1672609      6158       0.332276        1672609 = 1672609
  1699681      6210       0.332031        1699681 = 1699681
  1701001      6232        0.33306        1701001 = 31 * 37 * 1483
  1719481      6240       0.331441        1719481 = 19 * 90499
  1720489      6304       0.334729        1720489 = 97 * 17737
  1739929      6312       0.333015        1739929 = 83 * 20963
  1785169      6444       0.335045        1785169 = 149 * 11981
  1848289      6534       0.333069        1848289 = 1848289
  1851529      6604       0.336302        1851529 = 167 * 11087
  1906129      6768       0.338999        1906129 = 37 * 51517
  2031649      6964       0.336386        2031649 = 53 * 38333
  2144041      7062       0.330832        2144041 = 2144041
  2159161      7128       0.332592        2159161 = 1117 * 1933
  2213929      7156       0.329177        2213929 = 433 * 5113
  2222089      7226       0.331703        2222089 = 2222089
  2235601      7236       0.331019        2235601 = 569 * 3929
  2293729      7402       0.333709        2293729 = 2293729
  2356201      7408       0.328919        2356201 = 479 * 4919
  2413321      7604       0.333058        2413321 = 23 * 317 * 331
  2490049      7620       0.327878        2490049 = 23 * 108263
  2519449      7636       0.326384        2519449 = 73 * 34513
  2545321      7772       0.330275        2545321 = 41 * 62081
  2602681      7886       0.330907        2602681 = 2602681
  2636209      7892       0.328761        2636209 = 31 * 277 * 307
  2675401      7952       0.328497        2675401 = 281 * 9521
  2770681      8060       0.326412        2770681 = 53 * 61 * 857
  2784961      8234       0.332487        2784961 = 2784961
  2922529      8254       0.324302        2922529 = 2922529
  2938849      8348       0.326961        2938849 = 59 * 49811
  2959441      8468        0.33035        2959441 = 383 * 7727
  2985889      8548       0.331793        2985889 = 31 * 61 * 1579
  3100969      8596       0.326578        3100969 = 1319 * 2351
  3147769      8604       0.324118        3147769 = 1123 * 2803
  3197041      8664       0.323518        3197041 = 397 * 8053
  3199681      8708       0.325009        3199681 = 41 * 78041
  3206641      8788       0.327591        3206641 = 643 * 4987
  3267289      8810       0.324942        3267289 = 3267289
  3335929      9140       0.333166        3335929 = 101 * 33029
  3395401      9220       0.332734        3395401 = 709 * 4789
  3528001      9242       0.326369        3528001 = 3528001
  3581449      9356       0.327593        3581449 = 89 * 40241
  3674449      9412       0.324805        3674449 = 227 * 16187
  3712921      9628       0.330306        3712921 = 43 * 79 * 1093
  3818929      9818       0.331499        3818929 = 3818929
  3864169      9860       0.330706        3864169 = 37 * 181 * 577
  4064449      9904       0.322818        4064449 = 883 * 4603
  4217929      9910       0.316312        4217929 = 4217929
  4269841     10050       0.318569        4269841 = 4269841
  4270561     10092        0.31987        4270561 = 47 * 90863
  4275049     10150       0.321518        4275049 = 4275049
  4280329     10378        0.32851        4280329 = 4280329
  4499881     10420       0.320642        4499881 = 23 * 179 * 1093
  4586761     10660       0.324501        4586761 = 563 * 8147
  4701841     10770        0.32329        4701841 = 4701841
  4708729     10830       0.324822        4708729 = 4708729
  4724281     10932       0.327271        4724281 = 43 * 181 * 607
  4861081     11056       0.325688        4861081 = 859 * 5659
  4962049     11062       0.322102        4962049 = 4962049
  5013649     11082       0.320805        5013649 = 5013649
  5048761     11220       0.323522        5048761 = 37 * 136453
  5098129     11354       0.325591        5098129 = 5098129
  5343529     11410       0.318626        5343529 = 5343529
  5368609     11438       0.318564        5368609 = 5368609
  5390449     11566       0.321392        5390449 = 5390449
  5505361     11602       0.318577        5505361 = 5505361
  5542681     11684       0.319607        5542681 = 293 * 18917
  5587201     11744       0.319801        5587201 = 73 * 76537
  5634841     11984       0.324777        5634841 = 37 * 152293
  5730169     12104       0.324938        5730169 = 233 * 24593
  5815681     12308       0.327664        5815681 = 71 * 101 * 811
  6016921     12418       0.324309        6016921 = 6016921
  6080881     12430       0.322692        6080881 = 6080881
  6104641     12586       0.326024        6104641 = 6104641
  6195289     12648       0.324918        6195289 = 67 * 92467
  6406009     12912       0.325503        6406009 = 199 * 32191
  6589969     13026       0.323177        6589969 = 6589969
  6763849     13046       0.318957        6763849 = 6763849
  6783529     13194       0.322047        6783529 = 6783529
  6868801     13434       0.325605        6868801 = 6868801
  6969169     13504       0.324637        6969169 = 967 * 7207
  7209409     13856       0.326799        7209409 = 127 * 56767
  7478689     14180       0.327604        7478689 = 43 * 173923
  7623529     14206       0.324679        7623529 = 7623529
  7880161     14332       0.321509        7880161 = 47 * 167663
  7926361     14420        0.32242        7926361 = 359 * 22079
  8092681     14484       0.320087        8092681 = 79 * 89 * 1151
  8254801     14666       0.320511        8254801 = 8254801
  8281729     14724        0.32119        8281729 = 47 * 176207
  8465209     15118       0.325743        8465209 = 8465209
  8895769     15350       0.321638        8895769 = 8895769
  9033649     15706       0.326263        9033649 = 9033649
  9257329     15854       0.324838        9257329 = 9257329
  9850801     15906       0.314714        9850801 = 9850801
  9853561     15992       0.316366        9853561 = 173 * 56957
 10018849     16024       0.314049       10018849 = 47 * 59 * 3613
 10058161     16340       0.319538       10058161 = 41 * 245321
 10319761     16586       0.319702       10319761 = 10319761

=======================================================

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$\endgroup$
  • $\begingroup$ You have seven columns of numbers in that first table, and no heading on any of them. $\endgroup$ – Gerry Myerson Nov 20 '17 at 5:01

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