- Is it solvable in integers for any
*n*? - Ideal solutions
are known for
*n*= 1, 2, 3, 4, 5, 6, 7, 8 ,9, 11 and no other integers so far. - How to find new solutions for
*n*= 10 and*n >*= 12 ? - How to find the general solution for
*n*= 4, 5, 6, 7 ? - The general solution is known for ( k = 1 ) , ( k = 1, 2 ) , ( k = 1, 2, 3 ) only.
- The complete symmetric solution have been found for ( k = 1, 2, 3, 4 ) .
- How to find a new solution of the type ( k =1, 2, 3, 4, 5, 6, 7, 8 ) ?
- How to find non-symmetric ideal solutions of ( k =1, 2, 3, 4, 5, 6, 7, 8 ) and ( k =1, 2, 3, 4, 5, 6, 7, 8, 9 ) ?
- How to find a solution chain of the type ( k = 1, 2, 3, 4 ) ?
- [
*a*_{1},*a*_{2},*a*_{3},*a*_{4},*a*_{5}] = [*b*_{1},*b*_{2},*b*_{3},*b*_{4},*b*_{5}] = [*c*_{1},*c*_{2},*c*_{3},*c*_{4},*c*_{5}] ( k = 1, 2, 3, 4 ) - Solution chains are known for ( k = 1 ) , ( k = 1, 2 ) , ( k = 1, 2, 3 ) and ( k = 1, 2, 3, 4, 5 ) so far.
- Some other open problems are present on
`[44]`.

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

- Is (
*k*,*m*,*n*) always solvable when*m*+*n*>*k* - Is it true that (
*k*,*m*,*n*) is never solvable when*m*+*n*<*k* - For which
*k*,*m*,*n*such that*m*+*n*=*k*is (*k*,*m*,*n*) solvable ? - The 6 solvable cases with
*m*+*n*=*k*known at present are ( 4, 1, 3 ), ( 4, 2, 2 ), ( 5, 1, 4 ), ( 5, 2, 3 ), ( 6, 3, 3 ) and ( 8, 3, 5 ). - The only case with
*m*+*n*=*k*which

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

- T.N.Sinha conjectured that the above system has
nontrivial solution in positive integer for all
*n*. However he could only prove the cases*n<=*3. - With the results by Chen, we can prove that T.N.Sinha's
conjecture is true for
*n<=*4. See ( k = 2, 3, 4 ).

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

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

- [Question]: Is it always solvable in non-negative
integers for any (
*k*=*k*_{1}*k*_{2}*k*) ?_{n} - Non-negative solutions have been found to 105 types of (
*k*=*k*_{1}*k*_{2}*k*) so far. Among them, 39 types are all_{n}*k*> 0. ( See Non-negative integer solutions of the m=n+1 case ) - No any type has been proved to be unsolvable in non-negative integers so far.
- It is still unknown that whether there is non-negative
integer solution of any of the following types: ( k = 5 ) , ( k = 6 ) ,
( k = 1, 7 ) , ( k = 2, 8 ) or ( k = 1, 3, 9 ). All these types are the
*m*+*n*<*k*cases of the Questions by Lander-Parkin-Selfrige. - We may also consider the multigrade chains, the more general form of the equal sums of like powers system:
*c*_{11}^{k}+ c_{12}^{k}+ ... + c_{1m}^{k}= c_{21}^{k}+ c_{22}^{k}+ ... + c_{2m}^{k}*= ...... = c*_{j}_{1}^{k}+ c_{j}_{2}^{k}+ ... + c_{jm}^{k}- (
*k*=*k*_{1}*k*_{2}*k*)_{n} - Is it always solvable in non-negative integers
for any (
*k*=*k*_{1}*k*_{2}*k*) and any_{n}*j*? - Non-negative integer solution chains of
*j*>=3 have been found to 10 types. Among them, 7 types have been proved to be solvable for any*j*. ( See Multigrade Chains )

*a*_{1}^{h} +
a_{2}^{h} + ... + a_{n}^{h} = b_{1}^{h}
+ b_{2}* ^{h} + ... + b_{n}^{h}*
(

- [Question]: For which (
*h*=*h*_{1}*h*_{2}*h*) such that this system is solvable ?_{n} - When all
*h*> 0, integer solutions of the above system have been found for 14 types of (*h*=*h*_{1}*h*_{2}*h*) so far. ( See Integer solutions of the m=n case )_{n} - By appling Theorem
1, Theorem 3, Theorem
4 and Theorem 6, we can prove that
the following types of (
*h*=*h*_{1}*h*_{2}*h*) has no non-trivial integer solution:_{n} - (
*h*= 1, 2, ...,*n*) - (
*h*= 1, 3, ..., 2*n*-1 ) - (
*h*= 2, 4, ..., 2*n*)

- Conjecture I
- This conjecture is only for the cases of all
*k*> 0. - This conjecture is about the possible range of
each
*a*and_{i }*b*(_{i}*i*= 1, 2, ...,*n*+1 ). - This conjecture is useful for searching solutions of the above system by using computer.
- This conjecture will be announced later.
- Conjecture II
- Let 0 <=
*a*_{1}*<= a*_{2}<=*...*<=*a*_{n}_{+1 }, 0 <=*b*_{1}*<= b*_{2}<=*...*<=*b*_{n}_{+1 }, and*a*_{1}*<> b*_{1 }, then - (
*a*-_{i }*b*) (_{i }*a*_{i}_{+1 }-*b*_{i}_{+1 }) < 0 - This conjecture is for any case of (
*k*=*k*_{1}*k*_{2}*k*) ._{n} - This conjecture also can be stated as
- 0 <=
*a*_{1}*< b*_{1}*<= b*_{2}*< a*_{2}<=*a*_{3}*< b*_{3}*<= b*_{4}*< a*_{4 }<=*...*( assume*a*_{1}*< b*_{1 }) - P.Borwein
and C.Ingalls had found and proved the ( k = 1, 2, ...,
*n*) case of Conjecture II in 1994.`[44]` - When all
*k*> 0, Conjecture II is a corollary of Conjecture I. - Conjecture I and Conjecture II has been verified for all the known numerical examples by Chen Shuwen, and no counterexample is found.

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

Copyright 1997-2001, Chen Shuwen