1.Introduction

I found another parameter solution of A4+ B4+ C4 =  D4+ E4.

Proof is as very easy.

2.Theorem
      
      a,b:integer
     
　　　  The parameter solution of A4 + B4 + C4 = D4 + E4 exists.


　　  　A = 2b(a7 -a6b -2a5b2 +2a4b3 -2a3b4 -a2b5 +ab6 -b7)
        B = 3ab(a6 +a5b -a4b2 +2a3b3 +a2b4 -ab5+b6)
        C = a(2a7 +a6b -a5b2 +a4b3 -a3b4 -2a2b5 +2ab6 -2b7)
        D = a(2a7 +a6b -a5b2 +a4b3 +2a3b4 +a2b5 -ab6 +b7)
        E = -b(a7 +5a6b +a5b2 -a4b3 +4a3b4 +2a2b5 -2ab6 +2b7)
       
 　


     
Proof.

(a+x)4-(x)4-b4+(b+mx)4-(a-mx)4..............................................(1)
=(4a+4am3+4bm3)x3+(6a2+6b2m2-6a2m2)x2+(4a3+4a3m+4b3m)x


To do to make it to 0 the coefficient of x, m is decided. 
        m=-a3/(a3+b3)...................................................... (2)
Next,solve (4a+4am3+4bm3)x+(6a2+6b2m2-6a2m2)=0 about x


       x=3/2(-a2-b2m2+a2m2)/(a+am3+bm3)..................................... (3)

Substitute (2) to (3),and obtain x=3/2ab(a6+a5b-a4b2+2a3b3+a2b4-ab5+b6)/(a7-a6b-2a5b2+2a4b3-2a3b4-a2b5+ab6-b7)......................... (4)

Substitute (2) and (4) to (1),and obtain a parameter solution. 




Q.E.D.
　
                  
       
3.Example

I show a few examples.
a,b<=5,(a,b)=1


    
     (a,b)=(1,2)    2424 +    1894 +    1004 =     684 +    2634
     (a,b)=(1,3)   54874 +   27724 +   17304 =    9434 +   55864
     (a,b)=(1,4)  183084 +   68904 +   44714 =   23134 +  184144
     (a,b)=(1,5) 3333954 + 1001704 +  658844 =  334914 + 3341904
     (a,b)=(2,1)     134 +    2974 +    2904 =    3234 +    2514
     (a,b)=(2,3)   83194 +   91354 +   34584 =   35894 +  104074
     (a,b)=(2,5) 1119454 +  684954 +  406584 =  237174 + 1160654
     (a,b)=(3,1)   10734 +   42844 +   73504 =   75034 +   30584
     (a,b)=(3,2)   20144 +  129154 +   69424 =   98944 +  119774
     (a,b)=(3,4) 1010924 + 1294024 +  374254 =  535834 + 1394864
     (a,b)=(3,5) 4324154 + 4172404 + 1853344 = 1577914 + 5065304
     (a,b)=(4,1)   35374 +  100104 +  240044 =  241584 +   63194
     (a,b)=(4,3)  421174 + 1687144 +  625004 = 1125584 + 1607734
     (a,b)=(4,5) 6846054 + 9582304 + 2232044 = 4105464 + 10090854
     (a,b)=(5,1)  572294 + 1379704 + 4230204 = 4241154 +  796464
     (a,b)=(5,2)  181784 + 1083954 + 1473204 = 1538404 +  836974
     (a,b)=(5,3)  483514 + 6224404 + 4312904 = 5418554 + 5602264
     (a,b)=(5,4) 3595164 + 11850304 + 3343554 = 7356354 + 11432664