# Equal Sums of Like Powers

## Unsolved Problems and Conjectures

• The Prouhet-Tarry-Escott Problem (1851,1910)
•  Questions by Lander-Parkin-Selfrige (1967) 
• Conjecture by T.N.Sinha (1978)  
• a1k + a2k + ... + ank = b1k + b2k + ... + bnk      ( k = 1, 2, ..., j-1 , j+1, ..., n

• 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 ).
• Question I by Chen Shuwen (1997-2001)
• a1k + a2k + ... + an+1k = b1k + b2k + ... + bn+1k      ( k = k1 , k2 , ... , kn )

• [Question]: Is it always solvable in non-negative integers for any ( k = k1 , k2 , ... , kn ) ?
• Non-negative solutions have been found to 94 types of ( k = k1 , k2 , ... , kn ) so far. Among them, 39 types are all 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:
• c11k + c12k + ... + c1mk = c21k + c22k + ... + c2mk = ...... = cj1k + cj2k + ... + cjmk
( k = k1 , k2 , ..., kn )
• Is it always solvable in non-negative integers for any ( k = k1 , k2 , ... , kn ) and any  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 )
• Question II by Chen Shuwen (1997-2001)
• a1h + a2h + ... + anh = b1h + b2h + ... + bnh      ( h = h1 , h2 , ... , hn )

• [Question]: For which ( h = h1 , h2 , ... , hn ) such that this system is solvable ?
• By appling Theorem 1, Theorem 3, Theorem 4 and Theorem 6, we can prove that the following types of ( h = h1 , h2 , ... , hn ) has no non-trivial integer solution:
• ( h = 1, 2, ..., n )
• ( h = 1, 3, ..., 2n-1 )
• ( h = 2, 4, ..., 2n )
• Conjectures by Chen Shuwen (1997-2001)
• a1k + a2k + ... + an+1k = b1k + b2k + ... + bn+1k      ( k = k1 , k2 , ... , kn )

• Conjecture I
• This conjecture is only for the cases of all k > 0.
• This conjecture is about the possible range of each ai and bi ( 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 <= a1 <= a2 <= ... <= an+1 , 0 <= b1 <= b2 <= ... <= bn+1 , and a1 <> b1 , then
( ai - bi ) ( ai+1 - bi+1 ) < 0
• This conjecture is for any case of ( k = k1 , k2 , ... , kn ) .
• This conjecture also can be stated as
• 0 <= a1 < b1 <= b2 < a2 <= a3 < b3 <= b4 < a4 <= ...   ( assume a1 < b1 )
• P.Borwein and C.Ingalls had found and proved the ( k = 1, 2, ..., n ) case of Conjecture II in 1994. 
• 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.

Last revised May 6, 2001.
Copyright 1997-2001, Chen Shuwen