Equal Products and Equal Sums of Like Powers

a1a2 ... am = b1 b2 ... bm

a1k + a2k + ... + amk = b1k + b2k + ... + bmk      ( k = k2 , k3 , ... , kn )

• Introduction
• The Equal Products and Equal Sums of Like Powers is the system of the form
a1a2 ... am = b1 b2 ... bm
a1k + a2k + ... + amk = b1k + b2k + ... + bmk      ( k = k2 , k3 , ... , kn
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 , ... , am ] = [ b1 ,b2 , ... , bm       ( k = 0, k2 , k3 , ... , kn

• 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 Sr( A ) = Sr( B ):
a1a2 ... am = b1 b2 ... bm     <=>      S0( A ) = S0( B )
a1k + a2k + ... + amk = b1k + b2k + ... + bmk     <=>      Sk( A ) = Sk( B )
• We may also consider the graph of the following function to see more clearly:

y = { ( a1x+a2x+a3x+a4x+a5x ) / 5 }1/x - { ( b1x+b2x+b3x+b4x+b5x ) / 5 }1/x

 { ai } = { 54, 60, 63, 77, 80 } { bi } = { 56, 56, 66, 75, 81 } [ ai ] = [ bi ]      ( k = 0, 1, 2, 3 ) { ai } = { 1, 5, 9, 17, 18 } { bi } = { 2, 3, 11, 15, 19 } [ ai ] = [ bi ]      ( k = 1, 2, 3, 4 ) { ai } = { 1, 8, 13, 24, 27 } { bi } = { 3, 4, 17, 21, 28 } [ ai ] = [ bi ]      ( k = 1, 2, 3, 5 ) { ai } = { 7, 18, 55, 69, 81 } { bi } = { 9, 15, 61, 63, 82 } [ ai ] = [ bi ]      ( k = 1, 2, 3, 6 ) { ai } = { 3, 7, 10, 16, 16 } { bi } = { 4, 5, 12, 14, 17 } [ ai ] = [ bi ]      ( k = 1, 2, 4, 6 ) { ai } = { 3, 19, 37, 51, 53 } { bi } = { 9, 11, 43, 45, 55 } [ ai ] = [ bi ]      ( k = 1, 3, 5, 7 ) { ai } = { 71, 131, 180, 307, 308 } { bi } = { 99, 100, 188, 301, 313 } [ ai ] = [ bi ]      ( 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 , ... , kn ) has the same properties as the normal Equal Sums of Like Powers System ( k = k1, k2 , ... , kn ). 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 )

• General theorems
• Theorem 7 (By Chen Shuwen)
• If
[ a1 , a2 , ... , an ] = [ b1 , b2 , ... , bn
( k = 1, 2, ... , n-2, n
Then
[ Sa1 + T , Sa2 + T , ..., San + T ] = [ Sb1 + T , Sb2 + T , ... , Sbn + T ]
( k = 0, 1, 2, ... , n-2 )
where T = , S = .
• Theorem 8 (By Quo XianQiang)
• If
[ a1 , a2 , ... , am ] = [ b1 , b2 , ... , bm
( k = k1, k2 , ... , kn )
then
[ a1 , a2 , ... , am , Tb1 , Tb2 , ... , Tbm ] = [ b1 , b2 , ... , bm , Ta1 , Ta2 , ... , Tam
( k = 0, k1, k2 , ... , kn )
where T is arbitrary integer.

• 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 ... an+1 = b1 b2 ... bn+1
a1k + a2k + ... + an+1k = b1k + b2k + ... + bn+1k     ( k = k2 , ... , kn )
This system will be denoted as:
[ a1 , a2 , ... , an+1 ] = [ b1 ,b2 , ... , bn+1       ( k = 0, k2 , ... , kn )

When k1= 0 and all others k  > 0, idea non-negative integer solutions have been found to the following 24 types:

• ( 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 ]
• This type goes back at least to Bini, with partial solutions by Dubouis and Mathieu.[49]
• 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, 3 )
• [ 1, 108, 120 ] = [ 9, 10, 144 ]
• Stephane Vandemergel has found 62 solutions at least before 1994.[49]
• By Chen Shuwen in 2001.
• ( k = 0, 4 )
• [ 22, 93, 116 ] = [ 29, 66, 124 ]
• Stephane Vandemergel has found 3 solutions at least before 1994.[49]
• Found independently by Chen Shuwen in 2001.
• [ 3481, 21014, 21172 ] = [ 7847, 7906, 24964 ]
• By Chen Shuwen in 2001.
• ( 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, 3 )
• [ 5, 11, 16, 24 ] = [ 6, 8, 20, 22 ]
• First known solution, by Chen Shuwen in 2001.
• ( k = 0, 1, 4 )
• [ 7, 18, 60, 80 ] = [ 5, 36, 40, 84 ]
• There are only 1 solution in the range of max{ai,bi}<=150, by Chen Shuwen in 2017.
• ( k = 0, 1, 5 )
• [ 21, 31, 130, 143 ] = [ 13, 66, 91, 155 ]
• [ 91, 129, 196, 299 ] = [ 84, 161, 169, 301 ]
• There are only 2 solutions in the range of max{ai,bi}<=305, by Chen Shuwen in 2017.
• ( k = 0, 2, 3 )
• [ 855, 4338, 10406, 16335 ] = [ 1215, 2838, 11495, 15906 ]
• First known solution, by Chen Shuwen in 2022.
• ( k = 0, 2, 4 )
• [ 19, 67, 159, 207 ] = [ 23, 53, 171, 201 ]
• First known solution, by Chen Shuwen in 2001.
• ( k = 0, 2, 6 )
• [ 155, 779, 1455, 1887 ] = [ 185, 615, 1581, 1843 ]
• First known solution, by Chen Shuwen in 2017-05-19.
• ( 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, 4 )
• [ 4, 4, 15, 15, 22 ] = [ 3, 6, 11, 20, 20 ]
• [ 8, 9, 18, 21, 28 ] = [ 7, 12, 14, 24, 27 ]
• There are 21 solutions in the range of max{ai,bi}<=96, by Chen Shuwen in 2017.
• ( k = 0, 1, 2, 6 )
• [ 205, 248, 255, 344, 351 ] = [ 215, 221, 279, 328, 360 ]
• First known solution, by Chen Shuwen in 2017-05-31.
• ( k = 0, 2, 4, 6 )
• [ 2, 16, 25, 45, 48 ] = [ 3, 9, 32, 40, 50 ]
• [ 8, 33, 38, 68, 87 ] = [ 12, 17, 57, 58, 88 ]
• There are 10 solutions in the range of max{ai,bi}<=150, by Chen Shuwen in 2017.
• ( 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, 5 )
• [ 28, 57, 100, 174, 200, 245 ] = [ 30, 49, 125, 140, 228, 232 ]
• First known solution, by Chen Shuwen in 2017-04-15.
• ( k = 0, 1, 2, 4, 6 )
• [ 5, 12, 15, 32, 36, 46 ] = [ 6, 8, 23, 24, 40, 45 ]
• [ 8, 16, 24, 45, 55, 66 ] = [ 10, 11, 33, 36, 60, 64 ]
• There are 4 solutions in the range of max{ai,bi}<=100, by Chen Shuwen in 2017.
• ( 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, 6 )
• [ 32, 40, 41, 69, 72, 88, 90 ] = [ 33, 36, 45, 64, 80, 82, 92 ]
• [ 14, 33, 37, 108, 112, 192, 221 ] = [ 17, 21, 52, 88, 128, 189, 222 ]
• First known solution, by Chen Shuwen in 2017-05-26.
• ( k = 0, 1, 2, 4, 6, 8 )
• [ 437, 497, 732, 754, 1034, 1107, 1233 ] = [ 423, 548, 621, 902, 923, 1159, 1218 ]
• First known solution, by Chen Shuwen in 2023-02-12.
• ( 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.
• [ 11, 19, 22, 54, 60, 90, 92, 111 ] = [ 12, 15, 27, 46, 74, 76, 99, 110 ]
• [ 7, 15, 21, 50, 53, 88, 96, 110 ] = [ 8, 11, 32, 33, 70, 75, 105, 106 ]
• Smallest solution, by Chen Shuwen in 2017.
• ( k = 0, 1, 2, 3, 4, 5, 6, 7 )
• [ 387, 388, 416, 447, 494, 536, 573, 589, 610 ] = [ 382, 402, 403, 456, 485, 549, 559, 596, 608 ]
• First known solution, by Chen Shuwen in 2023-04-08.

• Ideal integer solutions
• Integer solutions have been found to the m=n case:
a1a2 ... an = b1 b2 ... bn
a1h + a2h + ... + anh = b1h + b2h + ... + bnh     ( h = h2 , ... , hn )
Also, this system will be denoted as:
[ a1 , a2 , ... , an ] = [ b1 ,b2 , ... , bn       ( h = 0, h2 , ... , hn )

When h1= 0 and all others h  > 0, idea integer solutions have been found to the following 6 types:

• ( h = 0, 1, 3 )
• [ -15, 1, 14] = [ -10, 3, 7 ]
• By A.Moessner in 1939. [13]
• ( h = 0, 1, 2, 4 )
• [ -138, 9, 62, 67 ] = [ -93, -23, -18, 134 ]
• By Ajai Choudhry in 2011. [1301]
• ( h = 0, 1, 3, 5 )
• [ -260, 32, 189, 323 ] = [ -210, 68, 114, 312 ]
• [ -555, -21, 646, 650 ] = [ -255, -78, 350, 703 ]
• By Chen Shuwen in 2017-03-26.
• [ -1518, 14, 1363, 1581 ] = [ -561, 138, 406, 1457 ]
• By Chen Shuwen in 2023-04-24.
• ( h = 0, 1, 2, 3, 5 )
• [ -21, -20, 2, 13, 26 ] = [ -26, -13, 5, 6, 28 ]
• By Ajai Choudhry in 2011. [1301]
• ( h = 0, 1, 2, 4, 6 )
• [ -48, -2, 16, 25, 45 ] = [ -40, -9, 3, 32, 50 ]
• By Chen Shuwen in 2017-03-21.
• ( h = 0, 1, 2, 3, 4, 6 )
• [ -11716, -6437, -1460, 1175, 7897, 10541 ] = [ -10865, -8383, -596, 3683, 4700, 11461 ]
• By Ajai Choudhry in 2011. [1301]

Last revised on May 2, 2023.