1.Introduction

A16+ A26+ A36+ A46+ A56+ A66 = B16+ B26+ B36+ B46+ B56+ B66.

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




2.Method


             A16+ A26+ A36+ A46+ A56+ A66 = B16+ B26+ B36+ B46+ B56+ B66.................(1)
                  

             A1=mqd-(p-q)c
             A2=(p+q)d-mpc
             A3=(p-q)d+mqc
             A4=mpd+(p+q)c
             A5=(up+vq)d-(vp-uq)c
             A6=(vp-uq)d+(up+vq)c
             B1=mqd+mpc
             B2=(p-q)d-(p+q)c
             B3=(p+q)d+(p-q)c
             B4=mpd-mqc
             B5=(vp-uq)d-(up+vq)c
             B6=(up+vq)d+(vp-uq)c
      

      A16+ A26+ A36+ A46+ A56+ A66 -( B16+ B26+ B36+ B46+ B56+ B66)
      =-3dc(q2+p2)(d2+c2)f

      f=(-2p4m5-4qp3u6-2qp3m6-8q3mp-4q4v5u-2qp3m5+2q2p2m5+8qp3m-4v5p4u+4u5p4v+4q4u5v
       +2q3m5p+4q3v6p-4qp3v6+4q3u6p+2q3m6p+28q2p2m+2q4m+20qp3v2u4+20qp3u2v4+24q2up2v5
       -24q2vp2u5-20q3v2pu4-20q3u2pv4-2q4m5+2p4m-16pq3+16p3q)d2
       +(10q3m4p-5p4m2-20p2q2-30q2m2p2+10p4-10qp3m4+5q2m6p2+10q4+20q3pm2-5q4m2-5m4p4-5q4m4-20qp3m2)cd
       +(2p4m5+4qp3u6+2qp3m6+8q3mp+4q4v5u+2qp3m5-2q2p2m5-8qp3m+4v5p4u-4u5p4v-4q4u5v
       -2q3m5p-4q3v6p+4qp3v6-4q3u6p-2q3m6p-28q2p2m-2q4m-20qp3v2u4-20qp3u2v4-24q2up2v5
       +24q2vp2u5+20q3v2pu4+20q3u2pv4+2q4m5-2p4m+16pq3-16p3q)c2
       = 0

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

     Discriminant=y2...................(2)


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

     For example,take m=1,u=1,v=2 then y2=25(576q4+3264pq3+3481p2q2-3264p3q+576p4)(2q+p)2(q-2p)2.

     This is the example which I described the last time.


  


3. Search result

         {m,u,v}<10, {p,q}<100
  
  m  u  v  p  q    A1    A2      A3    A4     A5     A6      B1    B2     B3     B4     B5     B6  

  1  0  2 51 46   2763   6595    584   1115   7274   4168   1699   2133   5531   3832   7664   3398
  1  2  3 47 22    752   5745  12151  29975  16414  66403  17824  11327  24321   6497  16157  66466
  1  3  1 46  7   1541    348   2225   4781   5813   8199   2556    667   4445   1889   5779   8223
  2  0  2  7 59   4500   2797   2899    145   4929    193   4747   1703   3044   1341   1341   4747
  2  1  2  1  4    179    232    350     87    244    347    210     53    263    316    211    368
  2  1  2 59 13  24958  66407  13176  32269  68500  22675  22356  41449  19093  60542  71720   7915
  2  1  3 82 61   4433    686   6064  13185   3747  19249  12240   5119   7121   2002   3117  19361
  2  1  5 69 25  12151  29975    752   5745  66403  16414  11327  17824   6497  24321  66466  16157
  2  2  1  1  4     30     11      7     16     32      9     28     19      9     10     33      4
  2  2  1  7 26    441    160    110    247    479    122    418    281    137    144    490     65
  2  2  1 11  4     55    112     34    117    159      8     26     57     83    140     44    153
  2  2  1 25  7    110    247    160    441    479    122    144    137    281    418     65    490
  2  2  1 29 19    522   1045    208    259    809    758     56    523    467    990    551    962
  2  2  1 79 31   1955   3942    869   3472   5119    778    616   1987   2603   4590   1679   4898
  2  2  2 77 39   1115    584   2763   6595   4168   7274   5531   1699   3832   2133   3398   7664
  2  4  2 53 39   2225   4781   1541    348   5813   8199    667   2556   1889   4445   5779   8223
  3  1  4 13 35   2460    129   1709   1722   2589   3431   2805   1396    474   1383   2779   3279
  3  1  4 62 65  17841   6612  18645  29059  11229  47704  35592  12589  11139   2175  14764  46731
  3  1  4 97 62    129   2460   1709   1722   2589   3431   2805   1396    474   1383   2779   3279
  3  2  2  4 11    535    729   1217    204    242   1282    375    319    569   1332    638   1138
  4  1  3  1  0      4     21      5     16     17     11     16      9      1     20     19      7
  4  1  5 55 93   6511   1488   5855   6028   5023  11883   9668   3165   1669   2992   6157  11337
  5  1  3  1  0      4     25      5     21     17     11     20      9      1     25     19      7
  5  5  1  1  2     23      9     28     19     41      3     35     11      3     20     39     13
  6  1  7 73 29  31592  81381  20243  47079 112973  26836  41913  21847   7876  89169 111016  34037
  6  1  7 85 79  38445  62633  71361  66848  24188 138209 115587  24020  14509  28533  52553 130096
  6  4  3  9  5    967   3084   1413    670   1654   2397   1962    809    155   2892   2754    947
  7  1  3  3  4     53    161    228     91     17    131    217     59      3    196    115     65
  7  1  8 12 47 253729  22567 223881 102530 231162 326411 290675  68671  14379 190022 258693 305054
  7  9  7  4  3    443    959    863    924   1271   2303   1603    257    201    196   2257   1351