1.Introduction

I found a parameter solution of A5+B5+C5+D5=E5+F5+G5
If I arranged the method which I used by the fourth-powers, 
I found out that I could apply it to the fifth-powers.
This time,I choose the variables so that the number of the items of the parameter solution
may decrease.



2.Theorem
      
      a,b:integer

　　  The parameter solution of A5 +B5 +C5 +D5 = E5 +F5 +G5 exists.


　      A=-(-b^3+a*b^2-a^2*b+a^3)^9*(b^2+a*b+a^2)*a^2*b*(4*a^30+66*b^30+151392*b^19*a^11
          -959762*b^14*a^16+515800*a^19*b^11+562407*a^21*b^9+635419*a^20*b^10-397009*a^17*b^13
          +152216*a^18*b^12-1317226*a^15*b^15+398164*a^22*b^8+232970*a^23*b^7+113369*a^24*b^6
          +45551*a^25*b^5+14790*a^26*b^4+3734*a^27*b^3+683*a^28*b^2+563047*b^22*a^8
          +515628*b^20*a^10+635795*b^21*a^9+233270*b^24*a^6+398664*b^23*a^7+45479*b^26*a^4
          +113385*b^25*a^5+3662*b^28*a^2+14666*b^27*a^3-398301*b^18*a^12-1317734*b^16*a^14
          -960846*b^17*a^13+639*a*b^29+78*a^29*b)

        B=(-b^5-a^2*b^3+a^3*b^2+a^5)^6*a*b^2*(-4*b^29+66*a^29+111*b*a^28+596*b^2*a^27
         +229*b^3*a^26+1470*b^4*a^25-1826*b^5*a^24+1024*b^6*a^23-7090*b^7*a^22
         +1466*b^8*a^21-7331*b^9*a^20+7476*b^10*a^19+1823*b^11*a^18+9302*b^12*a^17
         +5936*b^13*a^16-3424*b^14*a^15+148*b^15*a^14-13434*b^16*a^13-231*b^17*a^12
         -6660*b^18*a^11+4883*b^19*a^10+1690*b^20*a^9+4054*b^21*a^8+1376*b^22*a^7+190*b^23*a^6
         -514*b^24*a^5-709*b^25*a^4-388*b^26*a^3-183*b^27*a^2-46*b^28*a)

        C=2*(a+b)*(-b^3+a*b^2-a^2*b+a^3)^5*(-b^5-a^2*b^3+a^3*b^2+a^5)*a*b*(455*a^36*b^3+8*b^39
         +1387*a^34*b^5+22057*a^16*b^23+1515*a^33*b^6-2685*a^31*b^8-36212*a^20*b^19+862*a^22*b^17
         -20045*a^21*b^18-20045*a^18*b^21-36212*a^19*b^20+22057*a^23*b^16+26709*a^24*b^15
         +24262*a^25*b^14+9127*a^26*b^13-1000*a^27*b^12-10212*a^28*b^11-9687*a^29*b^10
         -7642*a^30*b^9-105*a^32*b^7+862*a^17*b^22+26709*a^15*b^24+985*a^4*b^35+1515*a^6*b^33
         +1387*a^5*b^34+178*a^2*b^37+455*a^3*b^36-105*a^7*b^32-2685*a^8*b^31-7642*a^9*b^30
         -9687*a^10*b^29-10212*a^11*b^28-1000*a^12*b^27+9127*a^13*b^26+24262*a^14*b^25+43*a*b^38
         +985*a^35*b^4+43*a^38*b+178*a^37*b^2+8*a^39)

        D=-2*b*a*(10*a^14+71*b*a^13+309*b^2*a^12+900*b^3*a^11+1973*b^4*a^10+3377*b^5*a^9
         +4620*b^6*a^8+5128*b^7*a^7+4620*b^8*a^6+3377*b^9*a^5+1973*b^10*a^4+900*b^11*a^3
         +309*b^12*a^2+71*b^13*a+10*b^14)*(-b^3+a*b^2-a^2*b+a^3)^9*(-b+a)*(b^2+a*b+a^2)^4
         *(-b^5-a^2*b^3+a^3*b^2+a^5)^2

        E=(-b^5-a^2*b^3+a^3*b^2+a^5)^6*a*b*(16*a^30-20*b^30+3436*b^19*a^11
         +10080*b^14*a^16-2620*a^19*b^11-5048*a^21*b^9+1725*a^20*b^10+756*a^17*b^13
         +11407*a^18*b^12-3424*a^15*b^15-2194*a^22*b^8-1856*a^23*b^7-290*a^24*b^6
         +548*a^25*b^5+549*a^26*b^4+420*a^27*b^3+191*a^28*b^2-842*b^22*a^8-4173*b^20*a^10
         +8204*b^21*a^9-1346*b^24*a^6+4256*b^23*a^7-1029*b^26*a^4+408*b^25*a^5-263*b^28*a^2
         -212*b^27*a^3-9815*b^18*a^12-3996*b^16*a^14-4888*b^17*a^13-36*a*b^29+56*a^29*b)

        F=-(-b^3+a*b^2-a^2*b+a^3)^9*(b^2+a*b+a^2)*a*b*(20*a^31+196*b*a^30+1211*b^2*a^29
         +5324*b^3*a^28+51821*b^5*a^26+121355*b^6*a^25+238740*b^7*a^24+395758*b^8*a^23
         +546005*b^9*a^22+605241*b^10*a^21+480118*b^11*a^20+127154*b^12*a^19-396621*b^13*a^18
         -928472*b^14*a^17-1265102*b^15*a^16-1265610*b^16*a^15-929556*b^17*a^14
         -397913*b^18*a^13+126330*b^19*a^12+479946*b^20*a^11+605617*b^21*a^10+546645*b^22*a^9
         +396258*b^23*a^8+239040*b^24*a^7+121371*b^25*a^6+51749*b^26*a^5+18316*b^27*a^4
         +5252*b^28*a^3+1167*b^29*a^2+184*b^30*a+18440*b^4*a^27+16*b^31)

        G=-2*b*a*(-b^5-a^2*b^3+a^3*b^2+a^5)^6*(2*a^30+2*b^30+416*b^19*a^11-1086*b^14*a^16
         +416*a^19*b^11-304*a^21*b^9+443*a^20*b^10+544*a^17*b^13-69*a^18*b^12+320*a^15*b^15
         +324*a^22*b^8-544*a^23*b^7+216*a^24*b^6-240*a^25*b^5+133*a^26*b^4-32*a^27*b^3
         +37*a^28*b^2+324*b^22*a^8+443*b^20*a^10-304*b^21*a^9+216*b^24*a^6-544*b^23*a^7
         +133*b^26*a^4-240*b^25*a^5+37*b^28*a^2-32*b^27*a^3-69*b^18*a^12-1086*b^16*a^14
         +544*b^17*a^13)

 　


     
Proof.

(a+n*x)^5+(b+m*x)^5+(a+b)^5+x^5 = (a+m*x)^5+(b+n*x)^5+(a+b+x)^5...........................(0)

=(5*a+5*b*n^4+5*a*m^4-5*a*n^4-5*b*m^4+5*b)*x^4
+(-10*b^2*m^3+10*a^2+10*b^2+10*a^2*m^3+20*a*b-10*a^2*n^3+10*b^2*n^3)*x^3
+(10*b^3+10*a^3*m^2-10*a^3*n^2+30*a*b^2+30*a^2*b+10*b^3*n^2-10*b^3*m^2+10*a^3)*x^2
+(5*a^4-5*a^4*n+20*a^3*b+30*a^2*b^2+5*b^4*n+5*b^4-5*b^4*m+5*a^4*m+20*a*b^3)*x



Decide m, n to make the coefficient of x and x^2 to 0.


       m=1/2*(2*b^4+3*a*b^3+9*a^2*b^2+5*a^3*b+5*a^4)*b/(b^3-a*b^2+a^2*b-a^3)/(b^2+a*b+a^2)

       n=-1/2*a*(5*b^4+5*a*b^3+9*a^2*b^2+3*a^3*b+2*a^4)/(b^5+a^2*b^3-a^3*b^2-a^5)

Next,solve (5*a+5*b*n^4+5*a*m^4-5*a*n^4-5*b*m^4+5*b)*x^4
     +(-10*b^2*m^3+10*a^2+10*b^2+10*a^2*m^3+20*a*b-10*a^2*n^3+10*b^2*n^3)*x^3=0 
about x


       x=-2*(2*a*b+b^2*n^3+a^2*m^3+a^2-a^2*n^3+b^2-b^2*m^3)/(-b*m^4+a+a*m^4+b-a*n^4+b*n^4)
       .......................... (1)

Substitute m,n to (1),and obtain 

       x=-(10*a^11+51*b*a^10+187*b^2*a^9+414*b^3*a^8+720*b^4*a^7+922*b^5*a^6
        +922*b^6*a^5+720*b^7*a^4+414*b^8*a^3+187*b^9*a^2+51*b^10*a+10*b^11)*(a^2+a*b+b^2)
        /(8*a^12+43*a^11*b+146*a^10*b^2+323*a^9*b^3+576*a^8*b^4+770*a^7*b^5+876*a^6*b^6
        +770*a^5*b^7+576*a^4*b^8+323*a^3*b^9+146*a^2*b^10+43*a*b^11+8*b^12)


Substitute m,n and x to (0),and obtain a parameter solution.


Q.E.D.　
 
　　　




　
                  
       
3.Example

I show a few examples.
a,b<=3


    

(2,-1)     130535+      53155+       5225+      45255 =      11785+     123975+      98405
(2,1)   232503115+   24271055+  212399115+   58118255 =  231771865+  213130365+   82389305
(3,-2) 6137416185+ 1731526555+ 1553055575+ 1538094355 = 1603357435+ 6087114325+ 3269620905
(3,-1)      97725+      11975+     481785+     548455 =     591475+     258705+     289755
(3,1)   250654075+   35636205+  195314575+  103729005 =  247179075+  198789575+  139365205