１．はじめに

I found a parameter solution of A5+B5+C5+D5=E5+F5+G5+H5.
This time too,I choose the variables so that the number of the items of the parameter solution
may decrease.


2.Theorem
      
      a,b:integer

　　 There is a parameter solution of A5 +B5 +C5 +D5 = E5 +F5 +G5 +H5.


　       A=15*a^29+129*b^29+388132*a^5*b^24-5188*a^3*b^26+11*a^28*b+663*a^27*b^2
          -5565064*a^17*b^12+4415334*a^15*b^14+7437661*a^13*b^16+1089508*a^14*b^15
          -15682310*a^12*b^17+7046504*a^10*b^19-4269188*a^19*b^10-2940205*a^16*b^13
          +152629*a^23*b^6-1614545*a^20*b^9+124948*a^21*b^8+2522952*a^11*b^18
          -7712413*a^9*b^20-4392452*a^7*b^22-8112668*a^18*b^11+15012*a^25*b^4
          +289942*a^22*b^7-1041*a*b^28+6953206*a^8*b^21+30244*a^24*b^5+906*a^26*b^3
          -124916*a^4*b^25-5827*a^2*b^27+775749*a^6*b^23


        B=-15*a^29-321*b^29+5139976*a^5*b^24+317422*a^3*b^26-103*a^28*b-875*a^27*b^2
         -400234*a^17*b^12-2548718*a^15*b^14-700997*a^13*b^16-3378116*a^14*b^15
         +4374148*a^12*b^17-22624268*a^10*b^19+2255228*a^19*b^10-2888947*a^16*b^13
         -42399*a^23*b^6+1368423*a^20*b^9+490696*a^21*b^8+6334376*a^11*b^18
         +30596163*a^9*b^20+22340116*a^7*b^22+2992184*a^18*b^11-14424*a^25*b^4
         +123202*a^22*b^7+6485*a*b^28-30073938*a^8*b^21-31916*a^24*b^5-3280*a^26*b^3
         -1532180*a^4*b^25-56289*a^2*b^27-12475767*a^6*b^23

        C=25*a^29+281*b^29-5188922*a^5*b^24-323604*a^3*b^26+113*a^28*b+1133*a^27*b^2
         -2859880*a^17*b^12-3596152*a^15*b^14+550719*a^13*b^16+217198*a^14*b^15
         -3129124*a^12*b^17+21760182*a^10*b^19-1366166*a^19*b^10-2829009*a^16*b^13
         +65063*a^23*b^6-515439*a^20*b^9-128326*a^21*b^8-5804102*a^11*b^18-31958571*a^9*b^20
         -22088978*a^7*b^22-3242562*a^18*b^11+16090*a^25*b^4-4220*a^22*b^7-6441*a*b^28
         +29784380*a^8*b^21+31450*a^24*b^5+3016*a^26*b^3+1464558*a^4*b^25+48701*a^2*b^27
         +12662395*a^6*b^23


        D=25*a^29-23*b^29-1139370*a^5*b^24-57068*a^3*b^26+33*a^28*b+781*a^27*b^2
         +7203240*a^17*b^12+1691992*a^15*b^14+13252671*a^13*b^16-14800002*a^14*b^15
         -7722348*a^12*b^17-17454058*a^10*b^19+5662234*a^19*b^10+586599*a^16*b^13
         -78521*a^23*b^6+1005545*a^20*b^9-265086*a^21*b^8+16360282*a^11*b^18
         +10085749*a^9*b^20+1824054*a^7*b^22+6266174*a^18*b^11+4482*a^25*b^4
         -319764*a^22*b^7-457*a*b^28-5127884*a^8*b^21-36182*a^24*b^5-944*a^26*b^3
         +375870*a^4*b^25+17781*a^2*b^27+909923*a^6*b^23


        E=15*a^29-23*b^29+2412908*a^5*b^24+128080*a^3*b^26-29*a^28*b+487*a^27*b^2
         -533504*a^17*b^12+7059406*a^15*b^14+13788637*a^13*b^16-6419092*a^14*b^15
         -17978922*a^12*b^17-12560616*a^10*b^19-754988*a^19*b^10-1232401*a^16*b^13
         +80837*a^23*b^6-854053*a^20*b^9+56568*a^21*b^8+13605144*a^11*b^18
         +13309747*a^9*b^20+7564064*a^7*b^22-3358300*a^18*b^11+9208*a^25*b^4
         +132170*a^22*b^7+1951*a*b^28-10502926*a^8*b^21-3572*a^24*b^5-1074*a^26*b^3
         -669260*a^4*b^25-21287*a^2*b^27-5100487*a^6*b^23


        F=25*a^29-321*b^29+5684320*a^5*b^24+332882*a^3*b^26+73*a^28*b+1105*a^27*b^2
         -2108038*a^17*b^12+4959882*a^15*b^14+1595615*a^13*b^16-9729092*a^14*b^15
         -6708044*a^12*b^17-43646428*a^10*b^19-2499140*a^19*b^10-5533019*a^16*b^13
         +115373*a^23*b^6-2145777*a^20*b^9-269796*a^21*b^8+25941496*a^11*b^18+48052295*a^9*b^20
         +28216352*a^7*b^22-2039376*a^18*b^11+19392*a^25*b^4+191582*a^22*b^7+6637*a*b^28
         -42030454*a^8*b^21+39876*a^24*b^5+2524*a^26*b^3-1665448*a^4*b^25-59281*a^2*b^27
         -14500543*a^6*b^23


        G=-15*a^29+281*b^29-5733266*a^5*b^24-339064*a^3*b^26-63*a^28*b-847*a^27*b^2
         -1152076*a^17*b^12-11104752*a^15*b^14-1745893*a^13*b^16+6568174*a^14*b^15
         +7953068*a^12*b^17+42782342*a^10*b^19+3388202*a^19*b^10-184937*a^16*b^13
         -92709*a^23*b^6+2998761*a^20*b^9+632166*a^21*b^8-25411222*a^11*b^18
         -49414703*a^9*b^20-27965214*a^7*b^22+1788998*a^18*b^11-17726*a^25*b^4
         -72600*a^22*b^7-6593*a*b^28+41740896*a^8*b^21-40342*a^24*b^5-2788*a^26*b^3
         +1597826*a^4*b^25+51693*a^2*b^27+14687171*a^6*b^23


　      H=25*a^29+129*b^29-3164146*a^5*b^24-190336*a^3*b^26+73*a^28*b+957*a^27*b^2
         +2171680*a^17*b^12-952080*a^15*b^14+6901695*a^13*b^16-7291402*a^14*b^15
         -5425736*a^12*b^17+2153062*a^10*b^19+2148034*a^19*b^10-1121205*a^16*b^13
         -6729*a^23*b^6+245053*a^20*b^9-196706*a^21*b^8+5278090*a^11*b^18-10936411*a^9*b^20
         -10132462*a^7*b^22+1511806*a^18*b^11+10286*a^25*b^4-161992*a^22*b^7-3449*a*b^28
         +12328248*a^8*b^21-2366*a^24*b^5+1036*a^26*b^3+920214*a^4*b^25+33241*a^2*b^27
         +6786159*a^6*b^23
 　　　　


     
Proof.

(a+m*x)^5+(b+n*x)^5+(a+b+x)^5+(a-b+x)^5-(a-b+m*x)^5-(a+b+n*x)^5-(b+x)^5-(a+x)^5

=(5*b*m^4-5*b+5*a-5*a*n^4)*x^4+(-20*a*b*n^3-10*a^2*n^3+20*a*b*m^3+10*a^2-10*b^2*m^3+10*b^2)*x^3
+(30*a^2*b*m^2+10*b^3*m^2-10*b^3-30*a*b^2*n^2-30*a*b^2*m^2+60*a*b^2-10*a^3*n^2-30*a^2*b*n^2
+10*a^3)*x^2+(-30*a^2*b^2*m-20*a*b^3*n+20*a^3*b*m+20*a*b^3*m+5*a^4-5*a^4*n-20*a^3*b*n
-30*a^2*b^2*n+5*b^4-5*b^4*m+60*a^2*b^2)*x............................................(0)


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


       m=(148*a^3*b^6-94*a^4*b^5-82*a^5*b^4+10*a^6*b^3-25*b^7*a^2-3*b^9+5*a^9+31*a^7*b^2
        +7*a^8*b-19*a*b^8)/(-61*b^7*a^2-68*a^4*b^5+120*a^6*b^3+60*a^5*b^4+5*a^8*b+61*a^7*b^2
        +3*a^9+64*a^3*b^6+37*a*b^8-3*b^9)


       n=(3*a^9+13*a^8*b+25*a^7*b^2+92*a^6*b^3+102*a^5*b^4+46*a^4*b^5-194*a^3*b^6
        +153*b^7*a^2-49*a*b^8+11*b^9)/(-61*b^7*a^2-68*a^4*b^5+120*a^6*b^3+60*a^5*b^4+5*a^8*b
        +61*a^7*b^2+3*a^9+64*a^3*b^6+37*a*b^8-3*b^9)


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

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

 
Substitute m,n to (1),and obtain 


       x=-1/4*(3708490*a^5*b^24+205796*a^3*b^26+103*a^28*b+1023*a^27*b^2-3879484*a^17*b^12
        +8460680*a^15*b^14-4605083*a^13*b^16+940426*a^14*b^15-5656456*a^12*b^17
        -23175222*a^10*b^19-6902402*a^19*b^10-1522867*a^16*b^13+15*a^29-129*b^29
        +164501*a^23*b^6-3759253*a^20*b^9-563786*a^21*b^8+14329030*a^11*b^18+28392543*a^9*b^20
        +16008698*a^7*b^22-6543366*a^18*b^11+23530*a^25*b^4+230372*a^22*b^7+3601*a*b^28
        -24284764*a^8*b^21+74158*a^24*b^5+4768*a^26*b^3-1053482*a^4*b^25-36233*a^2*b^27
        -8810935*a^6*b^23)/(10*a^28+38*b^28-33317*b^25*a^3+3865*b^26*a^2-748*b^27*a
        -2770548*b^17*a^11-426951*b^12*a^16-661018*b^13*a^15+1877150*b^14*a^14
        -1587744*b^15*a^13+574153*b^16*a^12+4901780*b^18*a^10-5255540*b^19*a^9
        +4364033*b^20*a^8-2989129*b^21*a^7+1469059*b^22*a^6-506194*b^23*a^5+136086*b^24*a^4
        -1188592*b^10*a^18+44*b*a^27+495*b^2*a^26+1451*b^3*a^25+8454*b^4*a^24+17948*b^5*a^23
        +17095*b^7*a^21-190123*b^8*a^20-878550*b^9*a^19-1257890*b^11*a^17+39443*b^6*a^22)



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)     6955753^5 + 36231625^5 + 23603111^5 + 46197943^5= 
              46197943^5 + 16280575^5 + 13636793^5 + 36873121^5

    (3,-1)    639358752^5 + 362934592^5 + 397628663^5 + 904670833^5=
              892879837^5 + 535190283^5 + 651149748^5 + 225372972^5

    (3,2)     1496802548597557^5 + 2868068559647757^5 + 4659519094582923^5 + 6416820889455797^5=
              3619962649518443^5 + 6515194318084677^5 + 3907625004712237^5 + 1398429119968677^5