[ eslpower.org ] Equal Sums of Like Powers

 

On the Generalization of the Prouhet-Tarry-Escott Problem

a1k + a2k + ... + amk = b1k + b2k + ... + bmk      ( k = k1 , k2 , ... , kn ) 

 


Equal Sums of Like Powers

Introduction


Equal Sums of Like Powers

Ideal non-negative integer solutions of

a1k + a2k + ... + an+1k = b1k + b2k + ... + bn+1k      ( k = k1 , k2 , ... , kn )

When m=n+1, non-negative integer solutions have been found to 42 + 42 + 24 + 23 + 33 = 164 types so far.
    Range of k
    Solved Types
    Reference
    all k  > 0
    42
    See below.
    all k  < 0
    42
    See Ideal positive integer solutions of k < 0 case
    k1= 0 and all others k  > 0
    24
    See Equal products and equal sums of like powers
    k1= 0 and all others k  < 0
    23
    See Ideal positive integer solutions of k < 0 case
    k1< 0 and kn > 0
    33
    See Ideal positive integer solutions of k < 0 case
When all k > 0 and m=n+1, idea non-negative integer solutions have been found to the following 42 types:


Equal Sums of Like Powers

Ideal integer solutions of

a1h + a2h + ... + anh = b1h + b2h + ... + bnh      ( h = h1 , h2 , ... , hn )

When m=n, no non-negative solution is found, but to the following 15 + 15 + 6 + 6 + 23 = 65 types, there are solutions in integer.
    Range of h
    Solved Types
    Reference
    all h  > 0
    15
    See below.
    all h  < 0
    15
    See Ideal integer solutions (h<0)
    h1= 0 and all others h  > 0
    6
    See Equal products and equal sums of like powers
    h1= 0 and all others h  < 0
    6
    See Ideal integer solutions (h<0)
    h1< 0 and hn > 0
    23
    See Ideal integer solutions (h<0)
When all h > 0 and m=n, idea integer solutions have been found to the following 15 types:

Equal Sums of Like Powers

Ideal prime solutions of

a1k + a2k + ... + an+1k = b1k + b2k + ... + bn+1k      ( k = k1 , k2 , ... , kn )

When m=n+1, ideal prime solutions have been found to the following 24 types:


Last revised on May 12, 2023.
 
Chen Shuwen
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