Equal Sums of Like Powers

Equal Products and Equal Sums of Like Powers

a1a2 ... am = b1 b2 ... bm

a1k + a2k + ... + amk = b1k + b2k + ... + bmk      ( k = k2 , k3 , ... , kn ) 


  • Introduction

  • Why use k = 0 for the equal products equation?
  • y = { ( a1x+a2x+a3x+a4x+a5x ) / 5 }1/x - { ( b1x+b2x+b3x+b4x+b5x ) / 5 }1/x

    { ai } = { 54, 60, 63, 77, 80 }
    { bi } = { 56, 56, 66, 75, 81 }
    [ ai ] = [ bi ]      ( k = 0, 1, 2, 3 )
    { ai } = { 1, 5, 9, 17, 18 }
    { bi } = { 2, 3, 11, 15, 19 }
    [ ai ] = [ bi ]      ( k = 1, 2, 3, 4 )
    { ai } = { 1, 8, 13, 24, 27 }
    { bi } = { 3, 4, 17, 21, 28 }
    [ ai ] = [ bi ]      ( k = 1, 2, 3, 5 )
    { ai } = { 7, 18, 55, 69, 81 }
    { bi } = { 9, 15, 61, 63, 82 }
    [ ai ] = [ bi ]      ( k = 1, 2, 3, 6 )
    { ai } = { 3, 7, 10, 16, 16 }
    { bi } = { 4, 5, 12, 14, 17 }
    [ ai ] = [ bi ]      ( k = 1, 2, 4, 6 )
    { ai } = { 3, 19, 37, 51, 53 }
    { bi } = { 9, 11, 43, 45, 55 }
    [ ai ] = [ bi ]      ( k = 1, 3, 5, 7 )
    { ai } = { 71, 131, 180, 307, 308 }
    { bi } = { 99, 100, 188, 301, 313 }
    [ ai ] = [ bi ]      ( k = 2, 4, 6, 8 )
    ( Ploted by Mathematica, Chen Shuwen )

  • General theorems

  • Ideal non-negative integer solutions

  • Ideal integer solutions

  • Last revised on May 31, 2017.
    Copyright 1997-2017, Chen Shuwen