Equal Products and Equal Sums of Like Powers

`Equal Sums of Like Powers`

Equal Products and Equal Sums of Like Powers

a₁a₂ ... a_m = b₁ b₂ ... b_m

a₁^k + a₂^k + ... + a_m^k = b₁^k + b₂^k + ... + b_m^k ( k = k₂ , k₃, ... , k_n )

Introduction
Why use k = 0 for the equal products equation?
Ideal non-negative integer solutions
Ideal integer solutions

Introduction

The Equal Products and Equal Sums of Like Powers is the system of the form

a₁a₂ ... a_m = b₁ b₂ ... b_m

a₁^k + a₂^k + ... + a_m^k = b₁^k + b₂^k + ... + b_m^k ( k = k₂ , k₃ , ... , k_n )

This system has been studied for a long time when k <= 3.[13] [5] [17] [6] [59]

In March of 2001, for the first time, Guo Xianqiang and Chen Shuwen found that the equal products equation can be regarded as the k = 0 case of the Equal Sums of Like Powers system.

In these pages, we will denote the Equal Products and Equal Sums of Like Powers system as:

[ a₁ , a₂ , ... , a_m ] = [ b₁ ,b₂ , ... , b_m ] ( k = 0, k₂ , k₃ , ... , k_n )

Why use k = 0 for the equal products equation?

The following explanation is given by Guo Xianqiang:

A similar expression are available on a web site about General Mean.

So both the equal products equation and the equal sums of like powers system can be expressed as S_r( A ) = S_r( B ):

a₁a₂ ... a_m = b₁ b₂ ... b_m <=> S₀( A ) = S₀( B )

a₁^k + a₂^k + ... + a_m^k = b₁^k + b₂^k + ... + b_m^k <=> S_k( A ) = S_k( B )

We may also consider the graph of the following function to see more clearly:

y = { ( a₁^x+a₂^x+a₃^x+a₄^x+a₅^x ) / 5 }^1/x - { ( b₁^x+b₂^x+b₃^x+b₄^x+b₅^x ) / 5 }^1/x

{ a_i } = { 54, 60, 63, 77, 80 } { b_i } = { 56, 56, 66, 75, 81 } [ a_i ] = [ b_i ] ( k = 0, 1, 2, 3 )	{ a_i } = { 1, 5, 9, 17, 18 } { b_i } = { 2, 3, 11, 15, 19 } [ a_i ] = [ b_i ] ( k = 1, 2, 3, 4 )
{ a_i } = { 1, 8, 13, 24, 27 } { b_i } = { 3, 4, 17, 21, 28 } [ a_i ] = [ b_i ] ( k = 1, 2, 3, 5 )	{ a_i } = { 7, 18, 55, 69, 81 } { b_i } = { 9, 15, 61, 63, 82 } [ a_i ] = [ b_i ] ( k = 1, 2, 3, 6 )
{ a_i } = { 3, 7, 10, 16, 16 } { b_i } = { 4, 5, 12, 14, 17 } [ a_i ] = [ b_i ] ( k = 1, 2, 4, 6 )	{ a_i } = { 3, 19, 37, 51, 53 } { b_i } = { 9, 11, 43, 45, 55 } [ a_i ] = [ b_i ] ( k = 1, 3, 5, 7 )
{ a_i } = { 71, 131, 180, 307, 308 } { b_i } = { 99, 100, 188, 301, 313 } [ a_i ] = [ b_i ] ( k = 2, 4, 6, 8 )	( Ploted by Mathematica, Chen Shuwen )

The Equal Products and Equal Sums of Like Powers System which we denote here as ( k = 0, k₂ , ... , k_n ) has the same properties as the normal Equal Sums of Like Powers System ( k = k₁, k₂ , ... , k_n ). For example:

Non-negative integer solutions are available only when m>=n+1.
Integer solutions are available only when m>=n.
Both satisfy the following relation ( See also Conjectures by Chen Shuwen ) when m=n+1:

0 <= a₁ < b₁ <= b₂ < a₂ <= a₃ < b₃ <= b₄ < a₄<= ... ( assume a₁ < b₁)

Ideal non-negative integer solutions

It seems that there is no non-negative integer solutions when m=n. Here we consider the m=n+1 case:

a₁a₂ ... a_n₊₁ = b₁ b₂ ... b_n₊₁

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

This system will be denoted as:

[ a₁ , a₂ , ... , a_n₊₁ ] = [ b₁ ,b₂ , ... , b_n₊₁ ] ( k = 0, k₂ , ... , k_n )

Non-negative integer solutions have been found to the following 8 types of this system:

( k = 0 )

[ 1, 6 ] = [ 2, 3 ]
[ 1, 24 ] = [ 2, 12 ] = [ 3, 8 ] = [ 4, 6 ]

( k = 0, 1 )

[ 4, 15, 15 ] = [ 5, 9, 20 ]

By A.Moessner in 1939.[13]

[ 2, 12, 15 ] = [ 3, 6, 20 ]

By A.Golden in 1944.[5]

( k = 0, 2 )

[ 13, 38, 51 ] = [ 17, 26, 57 ]

By G.Xeroudakes and A.Moessner in 1958.[17]

[ 15, 42, 48 ] = [ 16, 35, 54 ] = [ 21, 24, 60 ]

By T.N.Sinha in 1984.[6]

( k = 0, 1, 2 )

[ 2, 2, 11, 21 ] = [ 1, 6, 7, 22 ]

By A.Golden in 1944.[5]

[ 3, 14, 20, 40 ] = [ 4, 8, 30, 35 ]

By G.Xeroudakes and A.Moessner in 1958.[17]

( k = 0, 1, 2, 3 )

[ 4, 13, 17, 40, 50 ] = [ 5, 8, 25, 34, 52 ]

First known solution, by Guo Xianqiang in 2000.

[ 54, 60, 63, 77, 80 ] = [ 56, 56, 66, 75, 81 ]

By Chen Shuwen in 2001.

( k = 0, 1, 2, 3, 4 )

[ 169, 175, 192, 215, 216, 228 ] = [ 171, 172, 195, 208, 224, 225 ]

First known solution, by Chen Shuwen in 2001.

( k = 0, 1, 2, 3, 4, 5 )

[ 480, 497, 558, 595, 616, 663, 666 ] = [ 481, 495, 568, 578, 630, 651, 672 ]

First known solution, by Chen Shuwen in 2001.

( k = 0, 1, 2, 3, 4, 5, 6 )

[ 1899, 1953, 1957, 2079, 2117, 2231, 2241, 2323 ] = [ 1909, 1919, 2001, 2037, 2163, 2187, 2263, 2321 ]

First known solution, by Chen Shuwen in 2001.

Ideal integer solutions

Integer solutions have been found to the m=n case:

a₁a₂ ... a_n = b₁ b₂ ... b_n

a₁^h + a₂^h + ... + a_n^h = b₁^h + b₂^h + ... + b_n^h ( h = h₂ , ... , h_n )

Also, this system will be denoted as:

[ a₁ , a₂ , ... , a_n ] = [ b₁ ,b₂ , ... , b_n ] ( h = 0, h₂ , ... , h_n )

Only one type of this system has been solved until now:

( h = 0, 1, 3 )

[ -15, 1, 14] = [ -10, 3, 7 ]

By A.Moessner in 1939. [13]