1.Introduction


I show that there is a parameter solution for 6.4.4 by Euler's method.



2. Theorem


       There is a parameter solution for A16+ A26+ A36+ A46 = B16+ B26+ B36+ B46.

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



Proof.

             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.................(1)
    

      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=(40n2m4+4n2m6+100n2m2+m8+20n4m2+110m4+6n4m4-10n4+n8+4m2n6+20m6+25+100m2)p4
                 +(-240nm3-40nm5-240n3m-200nm-40n5m-80n3m3)qp3
                 +(-40n4m2-8m2n6+120n4-120m4-2m8+50+200n2-12n4m4-40m6-8n2m6-2n8-80n2m4)q2p2
                 +(40n5m+200nm+240n3m+40nm5+240nm3+80n3m3)q3p
                 +(40n2m4+4n2m6+100n2m2+m8+20n4m2+110m4+6n4m4-10n4+n8+4m2n6+20m6+25+100m2)q4

     Take n=2m,then the coefficient of p4 becomes to square number.
     
     (40n2m4+4n2m6+100n2m2+m8+20n4m2+110m4+6n4m4-10n4+n8+4m2n6+20m6+25+100m2)
     =(25+100m2+625m8+350m4+500m6)
     =52(1+2m2+5m4)2

      
     y2=(25+100m2+625m8+350m4+500m6)p4
        +(-2000m6-2400m4-400m2)qp3
        +(800m2-1000m6+50+1800m4-1250m8)q2p2
        +(400m2+2400m4+2000m6)q3p
        +(25+100m2+625m8+350m4+500m6)q4

     Take X=q/p,Y=y/(5p2) then
 
 
     Y2 =(1+4m2+25m8+14m4+20m6)X4
         +(16m2+96m4+80m6)X3
         +(32m2-40m6+2+72m4-50m8)X2
         +(-80m6-96m4-16m2)X
         +1+4m2+25m8+14m4+20m6

      1+4m2+25m8+14m4+20m6=(1+2m2+5m4)2

      We can obtain a rational solution by Euler's method.
      We must find rational numbers (h1,h2,h3).

     (h1X2+h2X+h3)2=(1+4m2+25m8+14m4+20m6)X4
                   +(16m2+96m4+80m6)X3
                   +(32m2-40m6+2+72m4-50m8)X2
                   +(-80m6-96m4-16m2)X
                   +1+4m2+25m8+14m4+20m6

     h3=1+2m2+5m4

     h2=1/2(-80m6-96m4-16m2)/(2h3)=1/2(-80m6-96m4-16m2)/(1+2m2+5m4)
       
     h1=(32m2-40m6+2+72m4-50m8-h22)/(2h3)
       =-(-1+1180m10+1000m14+650m12+625m16+728m8-82m4+16m6-20m2)/(1+2m2+5m4)3

     We get X using (h1,h2,h3).

     X=(16m2+96m4+80m6-2h1h2)/(h12-(1+4m2+25m8+14m4+20m6))

      =-8*(5m2+1)*(1+2m2+5m4)2/(625m12+375m10+500m8+470m6+97m4-13m2-6)

     We get (p,q) from X.

     q=-8(5m2+1)*(1+2m2+5m4)2
     p=(625m12+375m10+500m8+470m6+97m4-13m2-6)....................(2)

     Y=(1+2m2+5m4)(390625m24+468750m22+765625m20+962500m18+1083750m16+334500m14-76950m12
       +66000m10+125205m8+58910m6+13597m4+1692m2+100)
      /(5m2+3)2/(5m2-1)2/(25m8+5m6+21m4+11m2+2)2


     c=14+127m2+514m4+795m6-150m8-1675m10-250m12+625m14
     d=1875m14+1750m12+2475m10+1150m8+585m6+298m4+57m2+2..........(3)

     Substitute (2) and (3) to (1),then obtain A1,A2,A3,A4,B1,B2,B3,B4.

     A1=-100-228341m7-5848750m18-4453125m22+390625m26-1796875m24-6128125m20
        -135835m10-2089050m14-585450m12-4403250m16-68303m8-10409m4-35633m6
        -1544m2-180m-4440m3-42453m5-800723m9-1948455m11-3046050m13-2053250m15
        +4734375m21+4806250m19+1893750m17+2734375m23+1953125m27+390625m25

     A2=100-228341m7+5848750m18+4453125m22-390625m26+1796875m24+6128125m20
       +135835m10+2089050m14+585450m12+4403250m16+68303m8+10409m4+35633m6
       +1544m2-180m-4440m3-42453m5-800723m9-1948455m11-3046050m13-2053250m15
       +4734375m21+4806250m19+1893750m17+2734375m23+1953125m27+390625m25


     A3=-100+12147m7-1203750m18-3515625m22-1171875m26-2421875m24-1928125m20
        -899375m10-1918850m14-1565450m12-1705250m16-343479m8-14729m4-87077m6
        -1656m2-260m-3080m3-12089m5+305961m9+1362585m11+3419350m13+6376750m15
        +14921875m21+14756250m19+10578750m17+8359375m23+1953125m27+5078125m25


     A4=100+12147m7+1203750m18+3515625m22+1171875m26+2421875m24+1928125m20
       +899375m10+1918850m14+1565450m12+1705250m16+343479m8+14729m4+87077m6
       +1656m2-260m-3080m3-12089m5+305961m9+1362585m11+3419350m13+6376750m15
       +14921875m21+14756250m19+10578750m17+8359375m23+1953125m27+5078125m25


     B1=-100-300m-4403250m16-10409m4-135835m10-2089050m14-585450m12-68303m8
        -1544m2-35633m6-6640625m25+390625m26-1796875m24-4453125m22-6128125m20
        -5848750m18-1934585m11-755261m9-5926750m15-39867m5-209787m7-4856m3
        -3716350m13-9984375m21-8256250m19-7813750m17-11484375m23-1953125m27


     B2=100-300m+4403250m16+10409m4+135835m10+2089050m14+585450m12+68303m8
       +1544m2+35633m6-6640625m25-390625m26+1796875m24+4453125m22+6128125m20
       +5848750m18-1934585m11-755261m9-5926750m15-39867m5-209787m7-4856m3
       -3716350m13-9984375m21-8256250m19-7813750m17-11484375m23-1953125m27


     B3=-100-100m-1705250m16-14729m4-899375m10-1918850m14-1565450m12-343479m8
        -1656m2-87077m6-1171875m25-1171875m26-2421875m24-3515625m22-1928125m20
        -1203750m18+627705m11+206873m9-2259250m15-6089m5+15811m7-1432m3+394550m13
        -10328125m21-10493750m19-7101250m17-5390625m23+1953125m27


     B4=100-100m+1705250m16+14729m4+899375m10+1918850m14+1565450m12+343479m8
       +1656m2+87077m6-1171875m25+1171875m26+2421875m24+3515625m22+1928125m20
       +1203750m18+627705m11+206873m9-2259250m15-6089m5+15811m7-1432m3+394550m13
       -10328125m21-10493750m19-7101250m17-5390625m23+1953125m27




     For example,take m=2 then

    139947310345016+ 170841605289356+ 203404246908296+ 339913328839836
   =319359606279596+ 288465311335256+   6465951077096+ 142975033008636


    Q.E.D.


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