Equal Sums of Like Powers

Unsolved Problems and Conjectures

a₁^k + a₂^k + ... + a_n+1^k = b₁^k + b₂^k + ... + b_n+1^k      ( k = 1, 2, ..., n ) 

• 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 ) ?
• [ a₁, a₂, a₃, a₄, a₅ ] = [ b₁, b₂, b₃, b₄, b₅ ] = [ c₁, c₂, c₃, c₄, c₅ ]     ( 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].

a₁^k + a₂^k + ... + a_m^k = b₁^k + b₂^k + ... + b_n^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 has been proved to be unsolvable is ( 3, 1, 2 ).

a₁^k + a₂^k + ... + a_n^k = b₁^k + b₂^k + ... + b_n^k      ( 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]: Is it always solvable in non-negative integers for any ( k = k₁ , k₂ , ... , k_n ) ?
• Non-negative solutions have been found to 105 types of ( k = k₁ , k₂ , ... , k_n ) 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:
c₁₁^k + c₁₂^k + ... + c_1m^k = c₂₁^k + c₂₂^k + ... + c_2m^k = ...... = c_j1^k + c_j2^k + ... + c_jm^k 
     ( k = k₁ , k₂ , ..., k_n )
• Is it always solvable in non-negative integers for any ( k = k₁ , k₂ , ... , k_n ) 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]: For which ( h = h₁ , h₂ , ... , h_n ) such that this system is solvable ?
• When all h > 0, integer solutions of the above system have been found for 14 types of ( h = h₁ , h₂ , ... , h_n ) so far. ( See Integer solutions of the m=n case )
• By appling Theorem 1, Theorem 3, Theorem 4 and Theorem 6, we can prove that the following types of ( h = h₁ , h₂ , ... , h_n ) has no non-trivial integer solution:
• ( h = 1, 2, ..., n )
• ( h = 1, 3, ..., 2n-1 )
• ( h = 2, 4, ..., 2n )

a₁^k + a₂^k + ... + a_n+1^k = b₁^k + b₂^k + ... + b_n+1^k      ( k = k₁ , k₂ , ... , k_n )

• Conjecture I
• This conjecture is only for the cases of all k > 0.
• This conjecture is about the possible range of each a_i and 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 <= a₁ <= a₂ <= ... <= a_n+1 , 0 <= b₁ <= b₂ <= ... <= b_n+1 , and a₁ <> b₁ , then
( a_i - b_i ) ( a_i+1 - b_i+1 ) < 0
• This conjecture is for any case of ( k = k₁ , k₂ , ... , k_n ) .
• 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. [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.