*a*_{1}^{k}
+ a_{2}^{k} + ... + a_{m}^{k}
= b_{1}^{k} + b_{2}^{k}
+ ... + b_{n}^{k}

- 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 )

**( k, m, n ) = ( 3, 1, 2 )***a*^{3}*= b*^{3}*+ c*^{3}- 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 )**- 20615673
^{4 }= 2682440^{4 }+ 15365639^{4 }+ 18796760^{4 } - 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]` - 422481
^{4 }= 95800^{4 }+ 217519^{4 }+ 414560^{4 } - This is the smallest solution of this type, found
by Roger Frye.
`[45]` **( k, m, n ) = ( 4, 2, 2 )**- 59
^{4 }+ 158^{4 }= 133^{4 }+ 134^{4 } **( k, m, n ) = ( 5, 1, 4 )**- 144
^{5 }= 27^{5 }+ 84^{5 }+ 110^{5 }+ 133^{5 } - 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 )**- 14132
^{5}+ 220^{5}= 14068^{5}+ 6237^{5}+ 5027^{5} - 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 )**- 3
^{6}+ 19^{6}+ 22^{6}= 10^{6}+ 15^{6}+ 23^{6} - 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 )**- 966
^{8}+539^{8}+81^{8}= 954^{8}+ 725^{8}+ 481^{8}+ 310^{8}+158^{8} - By Scott Chase before 2000. ( See http://euler.free.fr/oldresults.htm )

- Please refer to Computing Minimal Equal Sums Of Like Powers.

Copyright 1997-2001, Chen Shuwen