1.Introduction

Choudhry(2000) obtained three solutions for following diophantine equation.

        x1n+ x2n+ x3n+ x4n = y1n+ y2n+ y3n+ y4n ,n=1,3,7


(i). 1741^k + 2435^k + 3004^k + 3476^k = 1937^k + 2111^k + 3280^k + 3328^k,  k=1,3,7
  

(ii). 1523^k + 4175^k + 4492^k + 5956^k = 1951^k + 3107^k + 5528^k + 5560^k,  k=1,3,7
    

(iii). 344^k + 902^k + 1112^k + 1555^k = 479^k + 662^k + 1237^k + 1535^k,  K=1,3,7
    


I show other solutions besides the Choudhry's one.


[1].Choudhry: EQUAL SUMS OF SEVENTH POWERS,ROCKY MOUNTAIN JOURNAL OF MATHEMATICS V30 (2000)


2. Method

x17+ x27+ x37+ x47 = y17+ y27+ y37+ y47......................................(1)


Set     x1=u1+v1+w1, x2=u1-v1-w1, x3=u2+v2-w2, x4=u2-v2+w2
        y1=u1+v1-w1, y2=u1-v1+w1, y3=u2+v2+w2, y4=u2-v2-w2 
        
Then (1) becomes to 
        
        u1v1w1{3(u14+v14+w14)+10(u12v12+v12w12+w12u12)}
       =u2v2w2{3(u24+v24+w24)+10(u22v22+v22w22+w22u22)}.......................(2)
   
To solve (2),set

       u1=p(x-a), v1=q(x-p), w1=b(x-q)
       u2=p(x-q), v2=q(x-a), w2=b(x-p)


Then (2) becomes to
        
        (12p4q-12q4p-20p3q2-20q3b2+12q4a-12b4q-20p2ab2+20q2ab2-12p4a+12b4p+20p3b2+20p2q3)x3
       +(8q4p2-8p4q2-18q4a2+18b4q2+10p2a2b2+40p2ab2q-10q2a2b2-40q2ab2p+18p4a2-18b4p2-40p3b2q-10p4b2+40p3aq2+40q3pb2+10q4b2-40p2q3a)x2
       +(-12q4p3-12b4q3-20p2a2b2q+20q2a2b2p-12p4a3+12b4p3+20p3b2q2+20p4b2q-20p3a2q2-20p4aq2+12p4q3+12q4a3+20q4p2a+20p2q3a2-20p2b2q3-20q4pb2)x
       +3p4a4-3b4p4-10p4b2q2+10p4a2q2+3b4q4-10q4p2a2+10p2b2q4-3q4a4=0.........(3)


If cubic equation (3) has a rational root,then diophantine equation (2) has a rational solution.

Finally,we obtain a integer solution of (1).

I searched the rational roots of cubic equation (3).




3. Results
         
         We obtained nine numerical solutions.

         2007.10.15: Add following solution.

         33704^k + 32317^k + 9803^k + 14977^k  = 33450^k + 32630^k + 10057^k + 14664^k








 
 
 
 
 
 

  No
  p
  q
  a
  b
  x
  k=1,3,7
  
  
  
  
  
  
  
  
  
  
 

 

  　
  47
  49
  57
  81
  1089/23
  1741^k + 2435^k + 3004^k + 3476^k =
  1937^k + 2111^k + 3280^k + 3328^k
  
 

 

  1
  49
  81
  41
  47
  975/23
  　
  
 

 

  　
  4
  19
  24
  111
  363/32
  　
  
 

 

  　
  　
  　
  　
  　
  　
  　
  
 

 

  2
  4
  15
  135
  81
  -261/4
  1523^k + 4175^k + 4492^k + 5956^k =
  1951^k + 3107^k + 5528^k + 5560^k
  
 

 

  　
  　
  　
  　
  　
  　
  　
  
 

 

  　
  39
  78
  151
  81
  4433/38
  3824^k + 5104^k + 2078^k + 1025^k =
  5129^k + 3635^k + 773^k + 2494^k
  
 

 

  3
  243
  486
  250
  234
  2040/7
  　
  
 

 

  　
  113
  226
  554
  454
  3164
  　
  
 

 

  　
  　
  　
  　
  　
  　
  　
  
 

 

  4
  63
  189
  20
  14
  -459/50
  5785^k + 5042^k + 7544^k + 2365^k =
  3583^k + 7244^k + 6742^k + 3167^k
  
 

 

  　
  34
  102
  759
  459
  -700
  　
  
 

 

  　
  225
  675
  -137
  50
  -459/14
  　
  
 

 

  　
  　
  　
  　
  　
  　
  　
  
 

 

  　
  94
  282
  217
  212
  235/4
  344^k + 902^k + 1112^k + 1555^k =
  479^k + 662^k + 1237^k + 1535^k
  
 

 

  　
  106
  318
  129
  141
  2544/5
  　
  
 

 

  5
  133
  399
  87
  63
  -3363/5
  　
  
 

 

  　
  51
  153
  223
  323
  247/3
  　
  
 

 

  　
  　
  　
  　
  　
  　
  　
  
 

 

  6
  49
  98
  264
  187
  24353/66
  58711^k + 42312^k + 9544^k +
  38285^k=16473^k + 56170^k + 51782^k + 24427^k
  
 

 

  　
  561
  1122
  694
  294
  297704/409
  　
  
 

 

  　
  　
  　
  　
  　
  　
  　
  
 

 

  7
  37
  74
  420
  304
  -9916/35
  76925^k
  + 52473^k + 50279^k + 15187^k=74895^k + 64419^k + 27857^k + 27693^k 
  
 

 

  　
  304
  608
  465
  74
  24385/59
  　
  
 

 

  　
  845
  1690
  2497
  1515
  45123/23
  　
  
 

 

  　
  　
  　
  　
  　
  　
  　
  
 

 

  8
  439
  878
  653
  247
  2111151/3634
  950437^k
  +656489^k +423012^k +81213^k=946714^k +688586^k +343064^k +132787^k  
  
  
  
 

 

  　
  　
  　
  　
  　
  　
  　
  
 

 

  9
  973
  1946
  2094
  14
  122320/81
  33704^k + 32317^k + 9803^k + 14977^k
  = 33450^k + 32630^k + 10057^k + 14664^k
  
 

 

  　
  127
  254
  -41
  313
  17907/140
  　
  
 

 

  　
  　
  　
  　
  　
  　
  　
  
 

 

  
  
  
  
 

 

  
  　
  Founded by Choudhry
  
  
 

 

  
  
  
  
 

 

  
  　
  Re-founded by Tomita
  
  
 

 

  
  
  
  
 

 

  
  No.3
  :Founded by Jaroslaw
  Wroblewski, 05/28/2002
  
 

 

  
  No.4
  :Founded by Jaroslaw
  Wroblewski, 05/28/2002
  
 

 

  
  No.6
  :Founded by Jaroslaw
  Wroblewski, 11/04/2006
  
 

 

  
  No.7
  :Founded by Jaroslaw
  Wroblewski, 11/04/2006
  
 

 

  
  No.8
  :Founded by Jaroslaw
  Wroblewski
  
 

 

  
  No.9
  :Founded by Jaroslaw
  Wroblewski, 11/04/2006


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