1.Introduction

I proved that there were always infinitely many parameter solutions of A4 +B4 +C4 +D4 +E4 =F4
in the last time.
For example,one of parameter solutions is
(2x2+36x-54)4+(2x2-36x-54)4+(4x2-108)4+(4x2+108)4+(3x2+81)4=(5x2+135)4

I forgot other types of parameter solutions just below.

(8s2+40st-24t2)4+(6s2-44st-18t2)4+(14s2-4st-42t2)4+(9s2+27t2)4+(4s2+12t2)4 =(15s2+45t2)4

I proved that other types of parameter solutions of A4 +B4 +C4 +D4 +E4 =F4 were in the infinity.



2.Theorem
      
     
    
    a1,a2,a3,a4,a5:integer

　　There are always infinitely many parameter solutions of A4 +B4 +C4 +D4 +E4 =F4.

    A=a1*x2+2*(a1+2*a2)*x-3*a1
    B=a2*x2-2*(a2+2*a1)*x-3*a2
    C=(a1+a2)*x2+(-2*a1+2*a2)*x-3*a1-3*a2
    D=a3*x2+3*a3
    E=a4*x2+3*a4
    F=a5*x2+3*a5


    Condition

    a54=a14+a24+(a1+a2)4+a34+a44

     
Proof.

    f=(a1*x2+2*(a1+2*a2)*x-3*a1)4+(a2*x2-2*(a2+2*a1)*x-3*a2)4+((a1+a2)*x2+(-2*a1+2*a2)*x-3*a1-3*a2)4+(a3*x2+3*a3)4+(a4*x2+3*a4)4-(a5*x2+3*a5)4

    Expanding and simplifying,then

　　f=(a34+4*a1*a23+a44+6*a12*a22+2*a14-a54+2*a24+4*a13*a2)*x8
     +(24*a14+48*a13*a2+24*a24+72*a12*a22+12*a34+48*a1*a23+12*a44-12*a54)*x6
     +(216*a13*a2+324*a12*a22+54*a34+108*a14+216*a1*a23+54*a44+108*a24-54*a54)*x4
     +(108*a34+648*a12*a22+432*a13*a2+216*a14+432*a1*a23-108*a54+216*a24+108*a44)*x2
     +324*a1*a23+486*a12*a22+324*a13*a2+81*a34-81*a54+81*a44+162*a24+162*a14

    Coefficient of x8=a34+4*a1*a23+a44+6*a12*a22+2*a14-a54+2*a24+4*a13*a2
    Let substitut a14+a24+(a1+a2)4+a34+a44 to a54,then we see that coefficient of x8=0.

    Coefficient of x6=24*a14+48*a13*a2+24*a24+72*a12*a22+12*a34+48*a1*a23+12*a44-12*a54

    Coefficient of x4=216*a13*a2+324*a12*a22+54*a34+108*a14+216*a1*a23+54*a44+108*a24-54*a54

    Coefficient of x2=108*a34+648*a12*a22+432*a13*a2+216*a14+432*a1*a23-108*a54+216*a24+108*a44

    Coefficient of x0=324*a1*a23+486*a12*a22+324*a13*a2+81*a34-81*a54+81*a44+162*a24+162*a14

    Similarly, the coefficient of x6,x4,x2 and x0 become 0. 
    Because all coefficients were adjusted to 0 by this, it is understood that the parameter 
    solution exists. 
    Moreover, because we can obtain the (a1,a2,a3,a4,a5) infinitely by identity of
    the Ramanujan's one(see below), there are infinitely many parameter solutions. 
    
    (8s2+40st-24t2)4+(6s2-44st-18t2)4+(14s2-4st-42t2)4+(9s2+27t2)4+(4s2+12t2)4 =(15s2+45t2)4

      (t,s)     a1     a2  a1+a2     a3    a4     a5
   
     ( 2,1)     84+  1544+  1624+  1174+   524=  1954 
     ( 3,1)    224+   724+   944+   634+   284=  1054
     ( 3,2)    564+  3464+  4024+  2794+  1244=  4654
     ( 4,1)   2164+  4584+  6744+  4414+  1964=  7354
     ( 4,3)    564+  1984+  2544+  1714+   764=  2854
     ( 5,1)    984+  1664+  2644+  1714+   764=  2854
     ( 5,2)   1684+  8664+ 10344+  7114+  3164= 11854
     ( 5,3)     64+   824+   884+   634+   284=  1054
     ( 5,4)   3284+  9064+ 12344+  8194+  3644= 13654
     ( 6,1)   6164+  9064+ 15224+  9814+  4364= 16354
     ( 6,5)   5364+ 12824+ 18184+ 11974+  5324= 19954
     ( 7,1)     64+    84+   144+    94+    44=   154
     ( 7,2)   5844+ 14744+ 20584+ 13594+  6044= 22654
     ( 7,3)    224+  1464+  1684+  1174+   524=  1954
     ( 7,4)    724+ 19464+ 20184+ 14674+  6524= 24454
     ( 7,5)   1064+  4624+  5684+  3874+  1724=  6454
     ( 7,6)   2644+  5744+  8384+  5494+  2444=  9154
     ( 8,1)  12084+ 14984+ 27064+ 17374+  7724= 28954
     ( 8,3)   1684+  7184+  8864+  6034+  2684= 10054
     ( 8,5)   2644+ 24984+ 27624+ 19534+  8684= 32554
     ( 8,7)  10964+ 22264+ 33224+ 21694+  9644= 36154
     ( 9,1)   3944+  4624+  8564+  5494+  2444=  9154
     ( 9,2)  11924+ 22264+ 34184+ 22234+  9884= 37054
     ( 9,4)   3764+ 29464+ 33224+ 23314+ 10364= 38854
     ( 9,5)    144+  8084+  8224+  6034+  2684= 10054
     ( 9,7)   2424+  7424+  9844+  6574+  2924= 10954
     ( 9,8)  14484+ 27944+ 42424+ 27634+ 12284= 46054
     (10,1)  19924+ 22344+ 42264+ 27094+ 12044= 45154
     (10,3)   3764+ 10224+ 13984+  9274+  4124= 15454
     (10,7)   7924+ 37944+ 45864+ 31414+ 13964= 52354
     (10,9)   6164+ 11424+ 17584+ 11434+  5084= 19054

   
Q.E.D.　
 
　　                  
       
3.Example

I show a few examples.

    

1.  64+    84+   144+    44+    94=   154
   (6x2+44x-18)4+(8x2-40x-24)4+(14x2+4x-42)4+(4x2+12)4+(9x2+27)4=(15x2+45)4 

2.  44+   224+   264+   214+   284=   354
   (4x2+96x-12)4+(22x2-60x-66)4+(26x2+36x-78)4+(21x2+63)4+(28x2+84)4=(35x2+105)4

3.  24+   444+   464+   394+   524=   654
   (2x2+180x-6)4+(44x2-96x-132)4+(46x2+84x-138)4+(39x2+117)4+(52x2+156)4=(65x2+195)4

4.  224+   524+   744+   574+   764=   954
   (22x2+252x-66)4+(52x2-192x-156)4+(74x2+60x-222)4+(57x2+171)4+(76x2+228)4=(95x2+285)4

5.  64+   824+   884+   284+   634=  1054
   (6x2+340x-18)4+(82x2-188x-246)4+(88x2+152x-264)4+(28x2+84)4+(63x2+189)4=(105x2+315)4

6.  224+   724+   944+   284+   634=  1054
   (22x2+332x-66)4+(72x2-232x-216)4+(94x2+100x-282)4+(28x2+84)4+(63x2+189)4=(105x2+315)4

7.  224+  1464+  1684+   524+  1174=  1954
   (22x2+628x-66)4+(146x2-380x-438)4+(168x2+248x-504)4+(52x2+156)4+(117x2+351)4=(195x2+585)4

8.  84+  1544+  1624+   524+  1174=  1954

   (8x2+632x-24)4+(154x2-340x-462)4+(162x2+292x-486)4+(52x2+156)4+(117x2+351)4=(195x2+585)4

9.  984+  1664+  2644+   764+  1714=  2854
   (98x2+860x-294)4+(166x2-724x-498)4+(264x2+136x-792)4+(76x2+228)4+(171x2+513)4=(285x2+855)4

10. 564+  1984+  2544+   764+  1714=  2854
   (56x2+904x-168)4+(198x2-620x-594)4+(254x2+284x-762)4+(76x2+228)4+(171x2+513)4=(285x2+855)4


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