dioph34e

1.Introduction

I found many parameter solutions for A₁⁷+ A₂⁷+ A₃⁷+ A₄⁷+ A₅⁷+ A₆⁷+ A₇⁷+ A₈⁷ = B₁⁷+ B₂⁷+ B₃⁷+ B₄⁷+ B₅⁷+ B₆⁷+ B₇⁷+ B₈⁷


2. Theorem
          
         There are many parameter solutions for the equation
         
         A₁⁷+ A₂⁷+ A₃⁷+ A₄⁷+ A₅⁷+ A₆⁷+ A₇⁷+ A₈⁷ = B₁⁷+ B₂⁷+ B₃⁷+ B₄⁷+ B₅⁷+ B₆⁷+ B₇⁷+ B₈⁷.




3. Proof

 (1+a1x)⁷+(1-a1x)⁷+(1+a2x)⁷+(1-a2x)⁷+(1+a3x)⁷+(1-a3x)⁷+(1+a4x)⁷+(1-a4x)⁷
-(1+b1x)⁷-(1-b1x)⁷-(1+b2x)⁷-(1-b2x)⁷-(1+b3x)⁷-(1-b3x)⁷-(1+b4x)⁷-(1-b4x)⁷

=(14a4⁶+14a1⁶+14a3⁶+14a2⁶-14b3⁶-14b1⁶-14b4⁶-14b2⁶)x⁶
+(70a2⁴+70a4⁴-70b2⁴+70a1⁴+70a3⁴-70b4⁴-70b3⁴-70b1⁴)x⁴
+(42a1²+42a4²-42b3²+42a2²-42b2²-42b1²+42a3²-42b4²)x²................(1)



It comes to solving the next simultaneous Diophantine equation.

     a1²+a2²+a3²+a4² = b1²+b2²+b3²+b4²

     a1⁴+a2⁴+a3⁴+a4⁴ = b1⁴+b2⁴+b3⁴+b4⁴

     a1⁶+a2⁶+a3⁶+a4⁶ = b1⁶+b2⁶+b3⁶+b4⁶

I found many solutions of the above simultaneous Diophantine equation. 

       a1>a2>a3>a4
       a1<100

      (a1, a2, a3, a4)    (b1, b2, b3, b4)

      ( 25  21  16   2)  ( 24  23  14   5)
      ( 34  25  24   7)  ( 32  31  15  14)
      ( 48  41  23   5)  ( 47  43  16  15)
      ( 50  36  31   7)  ( 49  41  20  18)
      ( 60  46  37   1)  ( 59  50  31  12)
      ( 61  46  40   3)  ( 59  53  30  16)
      ( 64  49  43  18)  ( 62  56  31  27)
      ( 67  49  47   9)  ( 63  61  31  23)
      ( 72  49  43   1)  ( 71  56  27  23)
      ( 72  62  31   1)  ( 71  64  23  18)
      ( 81  53  50   8)  ( 80  62  31  27)
      ( 83  56  51   2)  ( 82  64  37  21)
      ( 90  67  61   8)  ( 86  80  43  27)
      ( 93  70  64  23)  ( 89  83  42  40)
      ( 97  69  67   5)  ( 93  85  43  31)
      ( 98  79  48   5)  ( 97  82  40  21)

      For example,  (a1,a2,a3,a4,b1,b2,b3,b4)=(25, 21, 16, 2, 24, 23, 14, 5)

           25²+  21²+  16²+   2²  = 24²+  23²+  14²+   5²

           25⁴+  21⁴+  16⁴+   2⁴  = 24⁴+  23⁴+  14⁴+   5⁴

           25⁶+  21⁶+  16⁶+   2⁶  = 24⁶+  23⁶+  14⁶+   5⁶


      I think that above (a1,a2,a3,a4,b1,b2,b3,b4) exists in the infinity.

4. Example

         (1+25x)⁷+(1-25x)⁷+(1+21x)⁷+(1-21x)⁷+(1+16x)⁷+(1-16x)⁷+(1+2x)⁷+(1-2x)⁷ =
         (1+24x)⁷+(1-24x)⁷+(1+23x)⁷+(1-23x)⁷+(1+14x)⁷+(1-14x)⁷+(1+5x)⁷+(1-5x)⁷

         (1+34x)⁷+(1-34x)⁷+(1+25x)⁷+(1-25x)⁷+(1+24x)⁷+(1-24x)⁷+(1+7x)⁷+(1-7x)⁷ =
         (1+32x)⁷+(1-32x)⁷+(1+31x)⁷+(1-31x)⁷+(1+15x)⁷+(1-15x)⁷+(1+14x)⁷+(1-14x)⁷

         (1+48x)⁷+(1-48x)⁷+(1+41x)⁷+(1-41x)⁷+(1+23x)⁷+(1-23x)⁷+(1+5x)⁷+(1-5x)⁷ =
         (1+47x)⁷+(1-47x)⁷+(1+43x)⁷+(1-43x)⁷+(1+16x)⁷+(1-16x)⁷+(1+15x)⁷+(1-15x)⁷

         (1+50x)⁷+(1-50x)⁷+(1+36x)⁷+(1-36x)⁷+(1+31x)⁷+(1-31x)⁷+(1+7x)⁷+(1-7x)⁷ =
         (1+49x)⁷+(1-49x)⁷+(1+41x)⁷+(1-41x)⁷+(1+20x)⁷+(1-20x)⁷+(1+18x)⁷+(1-18x)⁷

         (1+60x)⁷+(1-60x)⁷+(1+46x)⁷+(1-46x)⁷+(1+37x)⁷+(1-37x)⁷+(1+x)⁷+(1-x)⁷ =
         (1+59x)⁷+(1-59x)⁷+(1+50x)⁷+(1-50x)⁷+(1+31x)⁷+(1-31x)⁷+(1+12x)⁷+(1-12x)⁷

         (1+61x)⁷+(1-61x)⁷+(1+46x)⁷+(1-46x)⁷+(1+40x)⁷+(1-40x)⁷+(1+3x)⁷+(1-3x)⁷ =
         (1+59x)⁷+(1-59x)⁷+(1+53x)⁷+(1-53x)⁷+(1+30x)⁷+(1-30x)⁷+(1+16x)⁷+(1-16x)⁷

         (1+64x)⁷+(1-64x)⁷+(1+49x)⁷+(1-49x)⁷+(1+43x)⁷+(1-43x)⁷+(1+18x)⁷+(1-18x)⁷ =
         (1+62x)⁷+(1-62x)⁷+(1+56x)⁷+(1-56x)⁷+(1+31x)⁷+(1-31x)⁷+(1+27x)⁷+(1-27x)⁷

         (1+67x)⁷+(1-67x)⁷+(1+49x)⁷+(1-49x)⁷+(1+47x)⁷+(1-47x)⁷+(1+9x)⁷+(1-9x)⁷ =
         (1+63x)⁷+(1-63x)⁷+(1+61x)⁷+(1-61x)⁷+(1+31x)⁷+(1-31x)⁷+(1+23x)⁷+(1-23x)⁷

         (1+72x)⁷+(1-72x)⁷+(1+49x)⁷+(1-49x)⁷+(1+43x)⁷+(1-43x)⁷+(1+x)⁷+(1-x)⁷ =
         (1+71x)⁷+(1-71x)⁷+(1+56x)⁷+(1-56x)⁷+(1+27x)⁷+(1-27x)⁷+(1+23x)⁷+(1-23x)⁷

         (1+72x)⁷+(1-72x)⁷+(1+62x)⁷+(1-62x)⁷+(1+31x)⁷+(1-31x)⁷+(1+x)⁷+(1-x)⁷ = 
         (1+71x)⁷+(1-71x)⁷+(1+64x)⁷+(1-64x)⁷+(1+23x)⁷+(1-23x)⁷+(1+18x)⁷+(1-18x)⁷

         (1+81x)⁷+(1-81x)⁷+(1+53x)⁷+(1-53x)⁷+(1+50x)⁷+(1-50x)⁷+(1+8x)⁷+(1-8x)⁷ =
         (1+80x)⁷+(1-80x)⁷+(1+62x)⁷+(1-62x)⁷+(1+31x)⁷+(1-31x)⁷+(1+27x)⁷+(1-27x)⁷

         (1+83x)⁷+(1-83x)⁷+(1+56x)⁷+(1-56x)⁷+(1+51x)⁷+(1-51x)⁷+(1+2x)⁷+(1-2x)⁷ =
         (1+82x)⁷+(1-82x)⁷+(1+64x)⁷+(1-64x)⁷+(1+37x)⁷+(1-37x)⁷+(1+21x)⁷+(1-21x)⁷

         (1+90x)⁷+(1-90x)⁷+(1+67x)⁷+(1-67x)⁷+(1+61x)⁷+(1-61x)⁷+(1+8x)⁷+(1-8x)⁷ =
         (1+86x)⁷+(1-86x)⁷+(1+80x)⁷+(1-80x)⁷+(1+43x)⁷+(1-43x)⁷+(1+27x)⁷+(1-27x)⁷

         (1+93x)⁷+(1-93x)⁷+(1+70x)⁷+(1-70x)⁷+(1+64x)⁷+(1-64x)⁷+(1+23x)⁷+(1-23x)⁷ =
         (1+89x)⁷+(1-89x)⁷+(1+83x)⁷+(1-83x)⁷+(1+42x)⁷+(1-42x)⁷+(1+40x)⁷+(1-40x)⁷

         (1+97x)⁷+(1-97x)⁷+(1+69x)⁷+(1-69x)⁷+(1+67x)⁷+(1-67x)⁷+(1+5x)⁷+(1-5x)⁷ =
         (1+93x)⁷+(1-93x)⁷+(1+85x)⁷+(1-85x)⁷+(1+43x)⁷+(1-43x)⁷+(1+31x)⁷+(1-31x)⁷

         (1+98x)⁷+(1-98x)⁷+(1+79x)⁷+(1-79x)⁷+(1+48x)⁷+(1-48x)⁷+(1+5x)⁷+(1-5x)⁷ =
         (1+97x)⁷+(1-97x)⁷+(1+82x)⁷+(1-82x)⁷+(1+40x)⁷+(1-40x)⁷+(1+21x)⁷+(1-21x)⁷
  



   Numerical example

  (1+25x)⁷-(1-24x)⁷-(1-23x)⁷+(1+21x)⁷+(1+16x)⁷-(1-14x)⁷-(1-5x)⁷+(1+2x)⁷ =
 -(1-25x)⁷+(1+24x)⁷+(1+23x)⁷-(1-21x)⁷-(1-16x)⁷+(1+14x)⁷+(1+5x)⁷-(1-2x)⁷

 
  1<=x<=20

  x       
                                               
  1  26⁷+  23⁷+  22⁷+  22⁷+  17⁷+  13⁷+   4⁷+   3⁷ =  24⁷+  25⁷+  24⁷+  20⁷+  15⁷+  15⁷+   6⁷+   1⁷ 
  2  51⁷+  47⁷+  45⁷+  43⁷+  33⁷+  27⁷+   9⁷+   5⁷ =  49⁷+  49⁷+  47⁷+  41⁷+  31⁷+  29⁷+  11⁷+   3⁷
  3  76⁷+  71⁷+  68⁷+  64⁷+  49⁷+  41⁷+  14⁷+   7⁷ =  74⁷+  73⁷+  70⁷+  62⁷+  47⁷+  43⁷+  16⁷+   5⁷
  4 101⁷+  95⁷+  91⁷+  85⁷+  65⁷+  55⁷+  19⁷+   9⁷ =  99⁷+  97⁷+  93⁷+  83⁷+  63⁷+  57⁷+  21⁷+   7⁷
  5 126⁷+ 119⁷+ 114⁷+ 106⁷+  81⁷+  69⁷+  24⁷+  11⁷ = 124⁷+ 121⁷+ 116⁷+ 104⁷+  79⁷+  71⁷+  26⁷+   9⁷
  6 151⁷+ 143⁷+ 137⁷+ 127⁷+  97⁷+  83⁷+  29⁷+  13⁷ = 149⁷+ 145⁷+ 139⁷+ 125⁷+  95⁷+  85⁷+  31⁷+  11⁷
  7 176⁷+ 167⁷+ 160⁷+ 148⁷+ 113⁷+  97⁷+  34⁷+  15⁷ = 174⁷+ 169⁷+ 162⁷+ 146⁷+ 111⁷+  99⁷+  36⁷+  13⁷
  8 201⁷+ 191⁷+ 183⁷+ 169⁷+ 129⁷+ 111⁷+  39⁷+  17⁷ = 199⁷+ 193⁷+ 185⁷+ 167⁷+ 127⁷+ 113⁷+  41⁷+  15⁷
  9 226⁷+ 215⁷+ 206⁷+ 190⁷+ 145⁷+ 125⁷+  44⁷+  19⁷ = 224⁷+ 217⁷+ 208⁷+ 188⁷+ 143⁷+ 127⁷+  46⁷+  17⁷
 10 251⁷+ 239⁷+ 229⁷+ 211⁷+ 161⁷+ 139⁷+  49⁷+  21⁷ = 249⁷+ 241⁷+ 231⁷+ 209⁷+ 159⁷+ 141⁷+  51⁷+  19⁷
 11 276⁷+ 263⁷+ 252⁷+ 232⁷+ 177⁷+ 153⁷+  54⁷+  23⁷ = 274⁷+ 265⁷+ 254⁷+ 230⁷+ 175⁷+ 155⁷+  56⁷+  21⁷
 12 301⁷+ 287⁷+ 275⁷+ 253⁷+ 193⁷+ 167⁷+  59⁷+  25⁷ = 299⁷+ 289⁷+ 277⁷+ 251⁷+ 191⁷+ 169⁷+  61⁷+  23⁷
 13 326⁷+ 311⁷+ 298⁷+ 274⁷+ 209⁷+ 181⁷+  64⁷+  27⁷ = 324⁷+ 313⁷+ 300⁷+ 272⁷+ 207⁷+ 183⁷+  66⁷+  25⁷
 14 351⁷+ 335⁷+ 321⁷+ 295⁷+ 225⁷+ 195⁷+  69⁷+  29⁷ = 349⁷+ 337⁷+ 323⁷+ 293⁷+ 223⁷+ 197⁷+  71⁷+  27⁷
 15 376⁷+ 359⁷+ 344⁷+ 316⁷+ 241⁷+ 209⁷+  74⁷+  31⁷ = 374⁷+ 361⁷+ 346⁷+ 314⁷+ 239⁷+ 211⁷+  76⁷+  29⁷
 16 401⁷+ 383⁷+ 367⁷+ 337⁷+ 257⁷+ 223⁷+  79⁷+  33⁷ = 399⁷+ 385⁷+ 369⁷+ 335⁷+ 255⁷+ 225⁷+  81⁷+  31⁷
 17 426⁷+ 407⁷+ 390⁷+ 358⁷+ 273⁷+ 237⁷+  84⁷+  35⁷ = 424⁷+ 409⁷+ 392⁷+ 356⁷+ 271⁷+ 239⁷+  86⁷+  33⁷
 18 451⁷+ 431⁷+ 413⁷+ 379⁷+ 289⁷+ 251⁷+  89⁷+  37⁷ = 449⁷+ 433⁷+ 415⁷+ 377⁷+ 287⁷+ 253⁷+  91⁷+  35⁷
 19 476⁷+ 455⁷+ 436⁷+ 400⁷+ 305⁷+ 265⁷+  94⁷+  39⁷ = 474⁷+ 457⁷+ 438⁷+ 398⁷+ 303⁷+ 267⁷+  96⁷+  37⁷
 20 501⁷+ 479⁷+ 459⁷+ 421⁷+ 321⁷+ 279⁷+  99⁷+  41⁷ = 499⁷+ 481⁷+ 461⁷+ 419⁷+ 319⁷+ 281⁷+ 101⁷+  39⁷
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