1.Introduction




I found many parameter solutions for A15+ A25+ A35+ A45+ A55+ A65 = B15+ B25+ B35+ B45+ B55+ B65.



2. Theorem
          
         There are infinitely many parameter solutions for 5.6.6 equation. 
         
          A15+ A25+ A35+ A45+ A55+ A65 = B15+ B25+ B35+ B45+ B55+ B65


          A1= 1+a1x
          A2= 1-a1x
          A3= 1+a2x
          A4= 1-a2x
          A5= 1+(a1+a2)x
          A6= 1-(a1+a2)x
          B1= 1+a4x
          B2= 1-a4x
          B3= 1+a5x 
          B4= 1-a5x
          B5= 1+(a4+a5)x
          B6= 1-(a4+a5)x

          a1=-10k-45n2-50-45mn+45k2-45m2+80n+85m
          a2=-90k-5n2+5-5mn-50k2-5m2+80kn+85km
          a4=30k-30n2+30+50mn+30k2+55m2-55km-95m
          a5=25k+55n2+25+60mn+25k2-25m2-55kn-95n

          k,m,n: integer







Proof.

(1+a1x)5+(1-a1x)5+(1+a2x)5+(1-a2x)5+(1+(a1+a2)x)5+(1-(a1+a2)x)5
-(1+a4x)5-(1-a4x)5-(1+a5x)5-(1-a5x)5-(1+(a4+a5)x)5-(1-(a4+a5)x)5........................(1)

=(-40a4a53+40a13a2-60a42a52+40a1a23+20a14+60a12a22+20a24-20a44-40a43a5-20a54)x4
+(-40a4a5-40a52+40a22+40a1a2+40a12-40a42)x2

Decide a1,a2,a4,a5 to make the coefficient of x2 and x4 to 0.

We obtain a parameter solution using one solution (a1,a2,a4,a5)=(45,5,30,25).

   a1=-10k-45n2-50-45mn+45k2-45m2+80n+85m
   a2=-90k-5n2+5-5mn-50k2-5m2+80kn+85km
   a4=30k-30n2+30+50mn+30k2+55m2-55km-95m
   a5=25k+55n2+25+60mn+25k2-25m2-55kn-95n...............................................(2)


Substitute (2) to (1),then obtain a parameter solution.



Q.E.D.

3. Example

(k m n)

(1 1 2)
(1-17x)5+(1+17x)5+(1+15x)5+(1-15x)5+(1-2x)5+(1+2x)5
=(1-5x)5+(1+5x)5+(1+18x)5+(1-18x)5+(1+13x)5+(1-13x)5

(1 2 3)
(1-23x)5+(1+23x)5+(1+9x)5+(1-9x)5+(1-14x)5+(1+14x)5
=(1+2x)5+(1-2x)5+(1+19x)5+(1-19x)5+(1+21x)5+(1-21x)5

(1 3 1)
(1-53x)5+(1+53x)5+(1+27x)5+(1-27x)5+(1-26x)5+(1+26x)5
=(1+51x)5+(1-51x)5+(1-13x)5+(1+13x)5+(1+38x)5+(1-38x)5

(1 3 2)
(1-91x)5+(1+91x)5+(1+37x)5+(1-37x)5+(1-54x)5+(1+54x)5
=(1+63x)5+(1-63x)5+(1+26x)5+(1-26x)5+(1+89x)5+(1-89x)5

(1 3 3)
(1-49x)5+(1+49x)5+(1+15x)5+(1-15x)5+(1-34x)5+(1+34x)5
=(1+21x)5+(1-21x)5+(1+29x)5+(1-29x)5+(1+50x)5+(1-50x)5

(2 2 3)
(1-67x)5+(1+67x)5+(1+70x)5+(1-70x)5+(1+3x)5+(1-3x)5
=(1+10x)5+(1-10x)5+(1+63x)5+(1-63x)5+(1+73x)5+(1-73x)5

(2 3 1)
(1-14x)5+(1+14x)5+(1+23x)5+(1-23x)5+(1+9x)5+(1-9x)5
=(1+21x)5+(1-21x)5+(1-2x)5+(1+2x)5+(1+19x)5+(1-19x)5

(2 3 2)
(1-11x)5+(1+11x)5+(1+12x)5+(1-12x)5+(1+x)5+(1-x)5
=(1+9x)5+(1-9x)5+(1+4x)5+(1-4x)5+(1+13x)5+(1-13x)5

(2 3 3)
(1-61x)5+(1+61x)5+(1+48x)5+(1-48x)5+(1-13x)5+(1+13x)5
=(1+27x)5+(1-27x)5+(1+37x)5+(1-37x)5+(1+64x)5+(1-64x)5

(3 1 1)
(1+71x)5+(1-71x)5+(1-47x)5+(1+47x)5+(1+24x)5+(1-24x)5
=(1+41x)5+(1-41x)5+(1+31x)5+(1-31x)5+(1+72x)5+(1-72x)5

(3 1 2)
(1+17x)5+(1-17x)5+(1-x)5+(1+x)5+(1+16x)5+(1-16x)5
=(1+11x)5+(1-11x)5+(1+8x)5+(1-8x)5+(1+19x)5+(1-19x)5

(3 2 1)
(1+13x)5+(1-13x)5+(1)5+(1)5+(1+13x)5+(1-13x)5
=(1+8x)5+(1-8x)5+(1+7x)5+(1-7x)5+(1+15x)5+(1-15x)5

(3 2 2)
(1+23x)5+(1-23x)5+(1+43x)5+(1-43x)5+(1+66x)5+(1-66x)5
=(1+34x)5+(1-34x)5+(1+33x)5+(1-33x)5+(1+67x)5+(1-67x)5

(3 3 2)
(1-23x)5+(1+23x)5+(1+87x)5+(1-87x)5+(1+64x)5+(1-64x)5
=(1+57x)5+(1-57x)5+(1+32x)5+(1-32x)5+(1+89x)5+(1-89x)5
 




  Numerical example  using next identity.
  
(1-11x)5+(1+11x)5+(1+12x)5+(1-12x)5+(1+x)5+(1-x)5
=(1+9x)5+(1-9x)5+(1+4x)5+(1-4x)5+(1+13x)5+(1-13x)5

  1<=x<=30
  x
  1   125+    85+  135+   35+   25+  125=  105+  105+   55+  115+  145
  2   235+   175+  255+   75+   35+  255=  195+  215+   95+  235+  275+   15
  3   345+   265+  375+  115+   45+  385=  285+  325+  135+  355+  405+   25
  4   455+   355+  495+  155+   55+  515=  375+  435+  175+  475+  535+   35
  5   565+   445+  615+  195+   65+  645=  465+  545+  215+  595+  665+   45
  6   675+   535+  735+  235+   75+  775=  555+  655+  255+  715+  795+   55
  7   785+   625+  855+  275+   85+  905=  645+  765+  295+  835+  925+   65
  8   895+   715+  975+  315+   95+ 1035=  735+  875+  335+  955+ 1055+   75
  9  1005+   805+ 1095+  355+  105+ 1165=  825+  985+  375+ 1075+ 1185+   85
 10  1115+   895+ 1215+  395+  115+ 1295=  915+ 1095+  415+ 1195+ 1315+   95
 11  1225+   985+ 1335+  435+  125+ 1425= 1005+ 1205+  455+ 1315+ 1445+  105
 12  1335+  1075+ 1455+  475+  135+ 1555= 1095+ 1315+  495+ 1435+ 1575+  115
 13  1445+  1165+ 1575+  515+  145+ 1685= 1185+ 1425+  535+ 1555+ 1705+  125
 14  1555+  1255+ 1695+  555+  155+ 1815= 1275+ 1535+  575+ 1675+ 1835+  135
 15  1665+  1345+ 1815+  595+  165+ 1945= 1365+ 1645+  615+ 1795+ 1965+  145
 16  1775+  1435+ 1935+  635+  175+ 2075= 1455+ 1755+  655+ 1915+ 2095+  155
 17  1885+  1525+ 2055+  675+  185+ 2205= 1545+ 1865+  695+ 2035+ 2225+  165
 18  1995+  1615+ 2175+  715+  195+ 2335= 1635+ 1975+  735+ 2155+ 2355+  175
 19  2105+  1705+ 2295+  755+  205+ 2465= 1725+ 2085+  775+ 2275+ 2485+  185
 20  2215+  1795+ 2415+  795+  215+ 2595= 1815+ 2195+  815+ 2395+ 2615+  195
 21  2325+  1885+ 2535+  835+  225+ 2725= 1905+ 2305+  855+ 2515+ 2745+  205
 22  2435+  1975+ 2655+  875+  235+ 2855= 1995+ 2415+  895+ 2635+ 2875+  215
 23  2545+  2065+ 2775+  915+  245+ 2985= 2085+ 2525+  935+ 2755+ 3005+  225
 24  2655+  2155+ 2895+  955+  255+ 3115= 2175+ 2635+  975+ 2875+ 3135+  235
 25  2765+  2245+ 3015+  995+  265+ 3245= 2265+ 2745+ 1015+ 2995+ 3265+  245
 26  2875+  2335+ 3135+ 1035+  275+ 3375= 2355+ 2855+ 1055+ 3115+ 3395+  255
 27  2985+  2425+ 3255+ 1075+  285+ 3505= 2445+ 2965+ 1095+ 3235+ 3525+  265
 28  3095+  2515+ 3375+ 1115+  295+ 3635= 2535+ 3075+ 1135+ 3355+ 3655+  275
 29  3205+  2605+ 3495+ 1155+  305+ 3765= 2625+ 3185+ 1175+ 3475+ 3785+  285
 30  3315+  2695+ 3615+ 1195+  315+ 3895= 2715+ 3295+ 1215+ 3595+ 3915+  295