Equal Sums of Like Powers

Non-negative Integer Solutions of

• Lander, Parkin and Selfridge found by computer the following triple coincidence of 4 sixth powers in 1966.[12]
1⁶ + 34⁶ + 49⁶ + 111⁶ = 7⁶ + 43⁶ + 69⁶ + 110⁶ = 18⁶ + 25⁶ +77⁶ + 109⁶
Chen Shuwen noticed in 1995 that
• [ 7, 43, 69, 110 ] = [ 18, 25, 77, 109 ]      ( k = 1, 2, 6 )
• Chen Shuwen found a parameter solution in 1997. Numerical examples are
• [ 31, 62, 107, 126 ] = [ 38, 51, 118, 119 ]
• [ 1, 80, 111, 148 ] = [ 5, 67, 124, 144 ]
• [ 67, 129, 138, 179 ] = [ 69, 118, 149, 177 ]
• Chen Shuwen searched the smallest solutions of this type by using a Pentium. There are solutions in the range of R<50. Here R = max { a_i, b_i }.
• [ 7, 16, 25, 30 ] = [ 8, 14, 27, 29 ]
• [ 15, 23, 27, 34 ] = [ 17, 19, 30, 33 ]
• [ 4, 26, 33, 45 ] = [ 6, 20, 39, 43 ]
• [ 5, 25, 31, 48 ] = [ 9, 16, 37, 47 ]