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]
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]