Equal Sums of Like Powers

Integer Solutions of

• Ajai Choudhry obtained solutions to this system in 1991. [59]
• [ -3254, 5583, 5658 ] = [ -1329, 2578, 6738 ]
• Chen Shuwen found the following 5-7-2-6 identity in 2017.
  • 9216 (a⁵ + b⁵ + c⁵ - d⁵ - e⁵ - f⁵ )( a⁷ + b⁷ + c⁷ - d⁷ - e⁷ - f⁷ ) = 35 ((a² + b² + c² - d² - e² - f² )³ -16 ( a⁶ + b⁶ + c⁶ - d⁶ - e⁶ - f⁶ ))²
    • where  a^h + b^h + c^h = d^h + e^h + f^h  ( h = 1, 3, 4 )
    • See Ramanujan 6-10-8 Identity and Hirschhorn 3-7-5 Identity for reference.[0901]
  • This ( Chen 5-7-2-6 identity) can also be expressed by defining
    • P₂ = (a² + b² + c² - d² - e² - f² ) / 2
    • P₅ = (a⁵ + b⁵ + c⁵ - d⁵ - e⁵ - f⁵ ) / 5
    • P₆ = (a⁶ + b⁶ + c⁶ - d⁶ - e⁶ - f⁶ ) / 6
    • P₇ = (a⁷ + b⁷ + c⁷ - d⁷ - e⁷ - f⁷ ) / 7
      • Then ( P₂³ / 12 - P₆ )² = P₅ P₇ , where a^h + b^h + c^h = d^h + e^h + f^h ( h = 1, 3, 4 )
  • Similarly, Ramanujan 6-10-8 Identity and Hirschhorn 3-7-5 Identity can be expressed as
    • R₃ = (a³ + b³ + c³ - d³ - e³ - f³ ) / 3
    • R₅ = (a⁵ + b⁵ + c⁵ - d⁵ - e⁵ - f⁵ ) / 5
    • R₆ = (a⁶ + b⁶ + c⁶ - d⁶ - e⁶ - f⁶ ) / 6
    • R₇ = (a⁷ + b⁷ + c⁷ - d⁷ - e⁷ - f⁷ ) / 7
    • R₈ = (a⁸ + b⁸ + c⁸ - d⁸ - e⁸ - f⁸ ) / 8
    • R₁₀ = (a¹⁰ + b¹⁰ + c¹⁰ - d¹⁰ - e¹⁰ - f¹⁰ ) / 10
      • Then  R₈ ² = 4/3 R₆ R₁₀  and  R₅ ² = R₃ R₇ , where a^h + b^h + c^h = d^h + e^h + f^h ( h = 1, 2, 4 )
      • Noted by Chen Shuwen, in 2017.