# Equal Sums of Like Powers

## Integer Solutions of

a1h + a2h + a3h = b1h + b2h + b3h
( h = 1, 3, 4)
• 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
• 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 ah + bh + ch = dh + eh + fh ( 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 ah + bh + ch = dh + eh + fh ( h = 1, 2, 4 )
• Noted by Chen Shuwen, in 2017.

Last revised Nov,5, 2017.