1.Introduction

I proved that there was always a parameter solution of A14+A24+A34+...+An4=B14+B24+B34+...+Bn4.



2.Theorem
      
     
    a1,a2,...,an:integer
    a1>a2>...>an>0
    n:integer

　　If n > 2, there is always a parameter solution of A14+A24+A34+...+An4=B14+B24+B34+...+Bn4.




     
Proof.

(mx+a1)4+(x+a2)4+...+(x+an)4=(mx-a2)4+(x-a1)4+...+(x-an)4............................(0)


coefficient of x=(4a13+4a23)m+4a13+4a23+8a33+...+8an3................................(1)

coefficient of x2=(6a12-6a22)m2+6a22-6a12............................................(2)

coefficient of x3=(4a1+4a2)m3+4a1+4a2+8a3+...+8an...................................(3)

4a13+4a23  is not to 0,so we obtain m from (1).

               m=(4a13+4a23+8a33+...+8an3)/(4a13+4a23)...............................(4)
From (2),(3),we obtain x.

               x=((6a12-6a22)m2+6a22-6a12)/((4a1+4a2)m3+4a1+4a2+8a3+...+8an).........(5)
               
Substitute (4) to (5).
Substitute m,x to (0),and obtain a parameter solution.
       
I omit each parameter solution because there is much number of the items.

   
Q.E.D.　
 
　　                  
       
3.Example

I show a few numerical examples.
a1,a2,a3,a4,a5<=6


    
Case n=4

(a1,a2,a3,a4)

(3 1 2 1):   23524+   64314+  82004+  64314=  94284+    6454+  11244+  28934

(4 1 2 1):  937134+  622364+ 595374+ 622364= 802184+  757314+ 703334+ 676344

(4 1 3 1):   66734+  190014+ 287934+ 190014= 311534+   54794+   5834+  92094

(4 1 3 2):   13064+   38214+  60034+  49124=  67614+   16344+   5434+   5484

(4 2 3 1):    2364+   19034+  23824+  14244=  26384+    9714+   4924+   4664

(4 2 3 2):  272594+ 1742904+2209894+1742904=2529354+ 1059044+ 592054+ 125064

(4 3 2 1):  167744+   29674+  47084+  64494=  45874+  151544+ 116724+  99314


Case n=5

(a1,a2,a3,a4,a5)

(5 1 4 3 2):     7404+    16014+   29394+   24934+   20474=
                34164+    10754+    6294+    1834+    2634

(5 2 4 3 1):   209204+  8150474+12171294+10160884+ 6140064=
             13863674+  5922404+ 3911994+ 1901584+ 2119244

(5 3 4 2 1):  2107904+  6698134+ 8098644+ 5297624+ 3897114=
              9096184+  4505954+ 3105444+  304424+ 1096094

(5 4 3 2 1):     1754+    11864+   10474+    9084+    7694= 
                14264+      654+    2134+    3524+    4914

(6 2 5 4 1):   188164+  2725344+ 4610874+ 3982364+ 2096834=
              5216244+  2302744+ 1674234+ 1045724+  839814

(6 2 5 4 3):    11764+   160424+  276314+  237684+  199054=
               320804+   148624+  109994+   71364+   32734

(6 4 3 2 1):   495664+   126974+  157444+  187914+  218384=
               190964+   431674+  340264+  309794+  279324

(6 4 5 3 2):    10264+    21804+   25714+   17894+   13984=
                28844+    17304+   13394+    5574+    1664

(6 5 4 3 2):    39364+    80214+   68264+   56314+   44364=
                92094+    51244+   27344+   15394+    3444