1.Introduction

A16+ A26+ A36+ A46 = B16+ B26+ B36+ B46.

This time,we generalize the last method and find the much smaller solutions.




2.Method


             A16+ A26+ A36+ A46 = B16+ B26+ B36+ B46.................(1)
       

             A1=(mq+np+q)d+(nq-mp+p)c
             A2=(mq+np-q)d+(nq-mp-p)c
             A3=(mp-nq-p)d+(mq+np+q)c
             A4=(mp-nq+p)d+(mq+np-q)c
             B1=(mq-np+q)d+(nq+mp+p)c
             B2=(mq-np-q)d+(nq+mp-p)c
             B3=(nq+mp-p)d+(np-mq+q)c
             B4=(nq+mp+p)d+(np-mq-q)c

      

      A16+ A26+ A36+ A46 -( B16+ B26+ B36+ B46)
      =-24qp(q2+p2)(d2+c2)(mc-nd)(md+nc)f

      f=(20cpqd-m4d2p2-m4c2q2+4m3dcq2n-4m3dcp2n+20m2cpqd-20mdcp2n+20mdcq2n+4mdcq2n3-4mdcp2n3+
         c2q2n4-c2p2n4+d2p2n4-d2q2n4-10m2c2q2+10m2c2p2+20dcqpn2+10m2d2q2+m4d2q2-10m2d2p2+5c2p2-5d2p2-
         5c2q2+5d2q2+m4c2p2)


     f=(-q2n4+10m2q2-m4p2-10m2p2-5p2+m4q2+5q2+p2n4)d2
      +(20pq-4m3p2n+20m2pq-20mp2n+4mq2n3-4mp2n3+4m3q2n+20qpn2+20mq2n)cd
      +(m4p2-10m2q2-m4q2+5p2-5q2+q2n4-p2n4+10m2p2)c2
      = 0

     We must find rational value (c,d) for above equation.

     Discriminant=(16m2n6+4n8+440m4-40n4+24m4n4+80m2n4+400m2n2+400m2+160m4n2+4m8+80m6+16m6n2+100)q4
                 +(160mn5+960nm3+320m3n3+960n3m+800nm+160m5n)pq3
                 +(-8n8-160m6+480n4-320m4n2-160m2n4-48m4n4-8m8-480m4-32m2n6+200-32m6n2+800n2)p2q2
                 +(-960n3m-800nm-160m5n-320m3n3-960nm3-160mn5)p3q
                 +(16m2n6+4n8+440m4-40n4+24m4n4+80m2n4+400m2n2+400m2+160m4n2+4m8+80*m6+16m6n2+100)p4
                 =y2...................(2)


     So, we must find rational numbers m,n,p,q,y.

     For example,take m=2,n=1,then y2=20800q4+19200pq3-27200p2q2-19200p3q+20800p4.
     This is the example which I described the last time.


  


3. Search result

         (m,n,p,q)<100
        

  m  n  p  q        A1        A2        A3        A4        B1        B2        B3        B4

  2  1  1  6       104        47       125        42       127        70        13        96
  2  1 13  1        16        25         2        21        23        14         5        24
  2  1 13 53      3164      1177      4250      1647      4282      2295       308      2911
  2  3  5  2       158       123         3        76        57        92        89       162
  2  3 22 13       481       342        24       161        69       208       301       486
  2  4  3 14      1561      1555      2433      1339      2197      2191       535      1629
  2  4  4  5       113        51       295       197       289       227        75        23
  2  4 11  8       313       195        97       139        93        25       257       299
  2  4 19 14      2177      1339       743      1005       733       105      1807      2069
  2  4 20 17      3479      1557      4223      3485      4359      2437      3475      2737
  2  4 29 26      7009      2667     11785      9043     11997      7655      7313      4571
  2  4 67 53      2433      1339      1555      1561      1629       535      2191      2197
  2  5  9  1       186       151        88       157       120       155       122       191
  2  6  3  5       777       655      2229      1595      2085      1963        49       585
  2  6  3 13      7695      8473     13539      9197     11859     12637      4505      8847
  2  8  9 16     30195     29353     95011     73369     87831     86989      7453     29095
  2  8 22  3    150245    124503     86261    118919     92019    117761    119297    151955
  2  9  6 11      4839      4888     15426     12245     14241     14290      1811      4992
  2  9 11 10       496       393       171       166       177        74       461       456
  2  9 65  6    439636    370791    286397    376734    303249    372094    354971    445308
  2 13  9 11       373       334      2229      1912      2119      2080        95       222
  2 16 47 92    173811    185753    561875    493753    527063    539005    129597    197719
  3  6 17 16       443       265       629       532       647       469       437       340
  3  6 20 21      2812      1319      8665      6803      8572      7079      2617       755
  3  6 33 25     10754      8197      1633      3020       953      1604      9058     10445
  3  6 47 36      8011      6092      1255      2261       820      1099      6763      7769
  3  6 51 64      2261      1255      8011      6092      7769      6763      1099       820
  3  6 52 25     63436     50915     13357     26351     13940     26461     50557     63551
  3  6 65 57     26242     18629     12065     12748     13177      5564     23522     24205
  4  1 23  7        65        78        11        46        50        37        46        81
  4  1 47  1      1030      1461       742      1415      1414       983       794      1467
  4  2 79 23      2615      2821       447      1741      1809      1603      1735      3029
  4  3  8  5        78        71        11        18        26        33        54        83
  4  3 12 47     33200     24249     51499     33108     49964     41013     14160     32551
  4  3 15 61      9673      7134     14789      9462     14338     11799      4182      9509
  4  3 31  8       207       206        50       141       134       135       138       229
  4  3 51 44     17249     20292     23912     13725     22888     19845     10716     20903
  4  3 60 59      2096     19023     68627     51156     68524     51597       816     18287
  4  3 89 27      4425      4334       837      2726      2606      2697      2970      4859
  4  5  4  1       209       188        72       139       115       136       153       220
  4  5 11 17        38        21       118        89       115        98         1        30
  4  5 44 39     18829     13348      7968      9631      9545      4064     16197     17860
  4  6 52 47     41391     29749     22147     23529     24959     13317     36999     38381
  4  7  7  8       355       164      1556      1257      1524      1333       220        79
  4  7  8 11       180       113       691       548       668       601        20       123
  4  7 16 27       384       295      1189       928      1136      1047        80       341
  4  7 41 47       909       422      3985      3218      3902      3415       554       213
  4  7 92 73      3985      3218       422       909       213       554      3415      3902
  4  7 96 91     17432     11565     22391     19928     23288     17421     16840     14377
  4  8 41 32    197671    162257     23191     43713      6447     28967    172439    192961
  4  8 68 77    162137     86175    734089    602447    716473    640511    106385     25257
  4 10 11 16       547       449      2279      1893      2175      2077        89       475
  4 11 19 31     15454     14051     57334     48087     54366     52963      6154     15401
  4 11 36 59      5608      5109     20687     17348     19612     19113      2264      5603
  4 11 53 32     14643     12622      2938      4249      2158      4179     13082     14393
  4 12  3 26     21941     23835     30851     25437     27749     29643     19485     24899
  4 21  1  4     12416     13203     22209     20036     20849     21636     11523     13696
  4 21 91 88   1808607   1644530    312610    319299    339958    175881   1732842   1739531
  5  6 79 76        60         7       295       249       297       244        67        21
  5 10 12 11      1519      1276       279       361       319        76      1379      1461
  5 10 37 64      9211      7973     30756     25631     29339     28101      4060      9185
  5 10 52 61       807       476      4463      3793      4343      4012       315       355
  5 10 91 74     63749     54456      6361     11654      1414      7879     57051     62344
  5 11 11  7     24439     21305      4215      6517      3259      6393     21841     24143
  6  1 29  9       502       573       156       337       339       268       417       598
  6  7 18 17      5040      3781      3751      3822      4158      2899      4584      4655
  6  7 59 18    402325    370476    148938    237055    196938    228787    331764    419881
  6 12 91 72   9952219   8753777    971971   1712057    245969   1444411   9040867   9780953
  7  6  5  4      1131      1012        47       153       124       243       957      1157
  7  6  8  3      3765      3553      1104      1937      1651      1863      3135      3968
  7  6 11  8      6283      5655        75      1216       911      1539      5307      6448
  7  6 16 47       383       315       744       595       719       651       243       392
  7  6 21 20      2983      3444      5803      4563      5564      5103      2337      3577
  7  6 33 10      6739      6444      2479      4047      3524      3819      5589      7157
  7  8  8 11       424       235      1920      1637      1888      1699        93       376
  7 14 11 52      6703      6863     10959      9404     10245     10405      5785      7340
  7 14 23 64      5707      5647     12479     10824     11825     11765      4545      6200
  7 14 39 38     12479     10824      5647      5707      6200      4545     11765     11825
  7 14 47 46     10959      9404      6863      6703      7340      5785     10405     10245
  7 14 58 55    162815    144051     23896     30557     28429      9665    151331    157992
  7 22 48 59     58649     50045    458200    420813    445673    437069     17517     54904
  8  3 27 19      6558      6419       258      1259      1650      1789      5518      7035
  8  6 11 58     10347      9193     15335     12165     14703     13549      7633     10803
  8 11 11 13        29        14       197       174       194       179         2        21
  8 11 44 43      8925      7004     11392     10805     11873      9952      8511      7924
  8 33 28 17   4490153   4254222   1020750   1142039    939579   1175510   4317373   4438662
  9  2  8 51      3453      2704      4615      3771      4589      3840      2627      3471
  9  2 44 27    112305    114244      7423     32295     34567     32628     97549    122421
  9  2 87 44     13637     14085      2316      5405      5329      4881     11908     14997
  9  6 23 82      3593      3025      6244      5165      6077      5509      2612      3691
  9  8  5 24     47616     43877     74024     61005     70771     67032     37120     50139
  9 17 67 39     98485     90987     21687     28589     21177     28675     91341     98243
 10  1 14  9      2528      2615       159       670       734       647      2256      2767
 10  8  5  8     11243      7329     41371     36497     41023     37109      6277     11151
 10  8 34 19     26571     25033      4795      8665      7021      8559     23567     27437
 10  8 41 16     78667     75521     25851     38513     33345     36491     69563     82225
 10  8 78 77      8245     13495     54203     47961     53843     48593      7089     13331
 10  8 88 85    887489    857163    481103    333829    422303    452629    784037    931311
 10  9 59 58       625      1098      5256      4675      5216      4743       486      1067
 10 13  7  2      3091      2928      1450      1835      1606      1769      2792      3177
 10 15 11 15      9249      6916     49971     45154     48959     46626      4179      8996
 10 25 83 85     33539     24106    295939    276756    291669    282236     31589     12406
 10 35  6  7       218       191      2381      2256      2336      2309        90       215
 10 35 17 13    738897    702178     91807    107588     74213    110932    713597    729378
 11  2 61 23       930       979       317       496       473       424       846      1025
 11  6 91 36      7639      7548      2447      3753      3372      3463      6803      8109
 11  8 43 56     11205      4618     64541     58258     64654     58067      5067     11350
 11  8 88 51    108814    103335     18382     33475     27498     32977     97381    112474
 11  8 89 88      2335      3262     10301      9138     10214      9287      2107      3270
 11 20 11 19      2304      2123      8383      7676      8156      7975      1725      2432
 12  1 27 40     95712     30929    415235    394392    431511    366728     82245    103088
 12  3  4  3       935       928         4       155       181       188       841       992
 12  3  7  8       211       346      3850      3617      3970      3413       446       679
 12  3 21  1      9158     10165      7714      9551      9301      8294      8593     10430
 12  3 37 15    180353    185938     57331     88718     83383     77798    164251    195638
 12 36 17 47  16042855  16270793  35263999  33280097  34228639  34456577  15014345  16998247
 12 36 17 55  24182897  24625375  47372249  44662855  45892985  46335463  22866271  25575665
 13  4  1 92      6836      6179      7152      5963      6977      6320      5820      7009
 14 22 15  8    139325    132251     37041     45311     37915     44989    131569    139839
 15  2 22 47       766       537      1913      1791      1954      1725       625       747
 15  6 34 33     33140     33037     13379      9055     12148     12251     30577     34901
 15  6 42 43       925      1811      8650      8087      8777      7891      1283      1846
 15 50 23  6     96511     93636     54177     57269     54300     57175     93519     96611
 16 22  2  3     22515     19289    104999     98093    103387    100161     16309     23215
 17 14  5  4      5421      5161       332       739       509       769      5076      5483
 17 14 59 48   2207360   2100321    116873    281585    189633    296672   2066969   2231681
 17 20  7  4      6470      6189      1502      1915      1601      1882      6114      6527
 17 22  1 44      3401      3437      3732      3379      3559      3595      3220      3573
 18  7 11  6      1413      1402       320       471       424       435      1322      1473
 18 26 86 83    148035    140485      3711      8849      4931      2619    141823    146961
 19 32 27 11    638369    616650    248950    278383    253738    275457    612401    641834
 21 13 33 61     35945     29763    113421    106483    113099    106917     29197     36135
 22 49 51 70     28714     26967    177709    171364    175543    173796     23821     30166
 23 34 52 55       391       191      7668      7361      7619      7419        23       284
 25 10 16 17        59        81      2123      2056      2155      2015       104       171
 27 10  4 79     11215     10689     12580     11577     12379     11853     10409     11412
 28 17 80 79   1369673   1336284    201484    125269    163687    197076   1312709   1388924
 28 24 17 16   4202595   4063109      9203    184725     83717    223203   4043379   4218901
 28 24 25 16  17872979  17412117   3395811   4259621   3683221   4144083  17191779  18055589
 29 22 97 29    250853    248496    126358    141749    133946    136303    241401    256792
 29 66  5  3    181690    177627     43391     46536     42840     46903    178109    181254
 31  5 63 31     72059     72893     21959     26757     25339     24505     69885     74683
 31 10 81 74   2460073   2423856     18749    126787    148208    184425   2364125   2509661
 36 58  3 17   1455749   1459815   2117187   2042881   2079653   2083719   1418269   1492575
 37  6 52 87   7869555   6422717  29004291  28375468  29376391  27929553   6978300   7607123
 37 48 45 79   5168906   4930131  18638111  18057282  18476621  18237846   4724322   5305151
 49 22 43 36    199300    196903     13275     20279     17119     19516    194463    201467
 52 55  8  3   3977508   3934247   1742981   1849316   1777796   1821057   3901220   4007555
 55 90 23 20  32240903  31745127   2073093   2425292   1967227   2463003  31819293  32171492
 57 34 84 89      3573      1156     96067     94515     96477     94060      2367      3919
 62 29 37 25  24844538  24665829   4401668   5116243   4732004   4910713  24385818  25100393
 71  8 56 79   3379008   2874775  18566024  18419827  18737176  18232943   3099963   3246160
 71 58 49 40  36349025  35912512   3301857   3962917   3456272   3892785  35795977  36457037
 72 63 28 27  39916010  39382369    352703    987082    505042   1038683  39330146  39964525