# Equal Sums of Like Powers

## Other Results on Equal Sums of Like Powers

a1k + a2k + ... + amk = b1k + b2k + ... + bnk

• m+n<k cases
• No integer solution is found when m+n<k.
• The special type of the m+n<k case is the well known Farmat's Last Theory.( m=1, n=2, k>3 )
• m+n=k cases
• ( k, m, n ) = ( 3, 1, 2 )
• a3 = b3 + c3
• This is the k=3 case of Farmat's Last Theory. Euler proved that there is no integer solution of this type first. [3]
• This is the unique m+n=k case of which the impossibility of solving has been proved.
• ( k, m, n ) = ( 4, 1, 3 )
• 206156734 = 26824404 + 153656394 + 187967604
• This solution is obtained by Noam D. Elkies, who solved this type first and thus disproved the n=4 case of Euler's generalization of Fermat's Last Throrem.[45]
• 4224814 = 958004 + 2175194 + 4145604
• This is the smallest solution of this type, found by Roger Frye. [45]
• ( k, m, n ) = ( 4, 2, 2 )
• 594 + 1584 = 1334 + 1344
• This equation was first studied by Euler. He gave a two-parameter solution in 1772.[9]
• See also ( k = 4 ) type for more.
• ( k, m, n ) = ( 5, 1, 4 )
• 1445 = 275 + 845 + 1105 + 1335
• This solution, obtained by L.J.Lander and Parkin, is the first known counterexample to Euler's conjecture on sums of like powers. [46]
• ( k, m, n ) = ( 5, 2, 3 )
• 141325 + 2205 = 140685 + 62375 + 50275
• Bob Scher and Ed Seidl obtained solution of this type first by using a week of computer time on a 72 node Intel Paragon in 1997. ( See Massively Parallel Number Theory )
• ( k, m, n ) = ( 6, 3, 3 )
• 36 + 196 + 226 = 106 + 156 + 236
• The first solution was obtained by Subba-Rao in 1934. [3] [31]
• See also ( k = 2, 6 ) type for more.
• ( k, m, n ) = ( 8, 3, 5 )
• m+n>k cases

• Last revised March,31, 2001.
Copyright 1997-2001, Chen Shuwen