单层等幂和数组
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.
- 4224814 = 958004 + 2175194
+ 4145604
- This is the smallest solution of this type, found
by Roger Frye.
- ( 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.
- 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.
- ( k, m, n ) = ( 5, 2,
3 )
- 141325 + 2205 = 140685
+ 62375 + 50275
平方和：3=1+2*2 , 9=7+4*2, 9=3+6*2, 9=1+8+4 ,11=9+6+2, 11=7+6*2, 13=5+12, 13=3+12+4, 7=3+6+2, ,