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Equal Sums of Like Powers

On the Integer Solutions of the Diophantine System

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 39 + 39 + 20 + 19 + 29 = 146 types so far.
    Range of k
    Solved Types
    Reference
    all k  > 0
    39
    See below.
    all k  < 0
    39
    See Ideal positive integer solutions of k < 0 case
    k1= 0 and all others k  > 0
    20
    See Equal products and equal sums of like powers
    k1= 0 and all others k  < 0
    19
    See Ideal positive integer solutions of k < 0 case
    k1< 0 and kn > 0
    29
    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 39 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 14 + 14 + 6 + 6 + 17 = 57 types, there are solutions in integer.
    Range of h
    Solved Types
    Reference
    all h  > 0
    14
    See below.
    all h  < 0
    14
    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
    17
    See Ideal integer solutions (h<0)
When all h > 0 and m=n, idea integer solutions have been found to the following 14 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 23 types:


Last revised on May 26, 2017.
 
Chen Shuwen
Seekway Technology Limited,
Building 3, Qunhua Hi-Tech Park,
15#, Qunhua Road,
Jiangmen City, Guangdong Province
People's Republic of China
 
Email: Chen.Shuwen@eslpower.org
Copyright 1997-2017, Chen Shuwen