Results and discussion on the k<0 cases

`Equal Sums of Like Powers`

Ideal integer solutions of h < 0 case

a₁^h + a₂^h + ... + a_n^h = b₁^h + b₂^h + ... + b_n^h ( h = h₁ , h₂ , ... , h_n with h₁ <0 )

Introduction

Ideal integer solutions ( all h < 0 )

Ideal integer solutions ( hn = 0 and all others h < 0 )

Ideal integer solutions ( h1 < 0 and hn > 0 )

Parametric solutions

Introduction

Please refer to the Introduction section of Ideal positive integer solutions of k < 0 case.
In this page, we will only consider the following system of h₁< 0 case and the ideal integer solutions ( m = n ):

a₁^h + a₂^h + ... + a_m^h = b₁^h + b₂^h + ... + b_m^h ( k = k₁ , k₂ , ... , k_n )

Ideal integer solutions ( all h< 0 )

When all h < 0, ideal integer solutions have been found to the following 15 types (all obtained by Chen Shuwen, using Theorem 9 ):

( h = -4, -2, -1 )

[ -99, 110, 990 ] = [ -90, 165, 198 ]
[ -257980650, 538394400,495322848 ] = [ -263469600, 825538080, 386970975 ] = [ -275179360, 1547883900, 334677600 ] = [ -287978400, 4127690400, 309576780 ]

( h = -6, -2, -1 )

[ -2028828511, 7333115100, 2884590300 ] = [ -4908456075, -3289992180, 2008741100 ]

( h = -4, -3, -1 )

[ -3376103253237849, 1967730608281562, 1941647222699887 ] = [ -8266245286708774, 4261380910021707, 1630430392703467 ]

( h = -4, -3, -2 )

[ -119374236504, 271759784220, 79485296365 ] = [ -125373714885, -237292689636, 78905111720 ]

( h = -5, -3, -2, -1 )

[ -15249, -106743, 320229, 13923 ] = [ -18837, -24633, 35581, 15249 ]
[ -939429685081041, -8046419476563699, -16824331632815007, 801158649181667 ] = [ -1033897474642263, -1989974709257689, 3776890774713573, 829899766641099 ] = [ -1242064751415873, -1350858744240621, 2682139825521233, 852846303967581 ]

( h = -6, -4, -2, -1 )

[ -8400, -8050, 13800, 38640 ] = [ -9200, -7728, 12075, 96600 ]

( h = -6, -4, -3, -2, -1 )

[ -8400, -19320, -38640, 13800, 8050 ] = [ -9200, -12075, 96600, 19320, 7728 ]

( h = -7, -5, -3, -2, -1 )

[ -15935205, -188035419, -940177095, 28490215, 16494335 ] = [ -17094129, -40877265, 72321315, 24107105, 18434845 ]

( h = -8, -6, -4, -2, -1 )

[ 1763005052031300, 955521822093300, -695407548301235, -407730810078900, 406407008747475 ] = [ -1264377360547700, -1251733586942223, 665815737735225, 415858334532300, -399914884007100 ]

( h = -7, -5, -4, -3, -2, -1 )

[ -929213535580, -1499412750595, -3298708051309, 2128198742780, 1783085433140, 984688970540 ] = [ -970208250385, -1244795491060, 65974161026180, 16493540256545, 1199530200476, 1081543623380 ]

( h = -8, -6, -4, -3, -2, -1 )

[ -13120911510, -4373637170, -2512514970, 1904648445, 2186818585, 3373948674 ] = [ -118088203590, -3191573070, -2811623895, 1935872190, 2071722870, 3936273453 ]
- First known solution, by Jaroslaw Wroblewski and Chen Shuwen in 2019.

( h = -9, -7, -5, -3, -2, -1 )

[ -3026260512063434639, -3872986251189991481, -12669260109824887387, 8214135675600751163, 2586457946296430297, 2314199215107332371 ] = [ -2778759652340774557, -4320730326472071421, -106783763782809765119, 25775391257919598477, 2403493075497325903, 2388135292267311041 ]

( h = -10, -8, -6, -4, -2, -1 )

[ -851747928479140, 4916908496220490, 1773311260931980, 1257813801358730, 772657049406077, 716370774283780 ] = [ -1150765818264370,-3090628197624308, 2301531636528740, 893983362949180, 740904019978430, 730891803492235 ]

( h = -8, -6, -5, -4, -3, -2, -1 )

[ -19661789772129923000, -33658318084493597000, -35044248829149215700, 126755793637348227000, 66938452819723221000, 23830089203821466676, 22566372352103661625 ] = [ -20126764530254617125, -26360718145820206500,-64059379580165233000, -198584076698512222300, 34436545092227553000, 31858408026499287000, 20903587020896023400 ]

( h = -9, -7, -6, -5, -4, -3, -2, -1 )

[ -15149134, -16526328, -31615584, -103879776, 51939888, 31615584, 18645088, 15807792 ] = [ -14839968, -17313296, -27967632, 727158432, 181789608, 22723701, 22035104, 15471456, ]

Ideal integer solutions ( h_n= 0 and all others h< 0 )

When h₁= 0 and all others h< 0, integer solutions have been found to the following 6 types (all obtained by Chen Shuwen, using Theorem 9 ):

( h = -3, -1, 0 )

[ [ -21, 30, 70 ] = [ -14, 15， 210 ]

( h = -4, -2, -1, 0 )
- [ -47771, -37386, -9246, 6417 ] = [ -6231, 12834, 13869, 95542 ]
( h = -5, -3, -1, 0 )
- [ -604656, 406980, 1113840, 1867320 ] = [-488376, 393120, 671840, 3968055 ]
( h = -6, -4, -2, -1, 0 )
- [ -3600, -150, 160, 450 ] = [ -800, -180, 144, 225, 2400 ]
( h = -5, -3, -2, -1, 0 )
- [ -273, -260, 210, 420, 2730 ] = [ -420, -210, 195, 910, 1092 ]
( h = -6, -4, -3, -2, -1, 0 )
- [ -368820196108882435, -83653485524152300, -45960864315377975, 51084098882361100, 68187601154738300, 458278711760824132 ] = [ -49560744253931740, -64234460970889700, 46983464472469100, 114569677940206033, 146206214042619700, 903485715300282475 ]

Ideal integer solutions ( h₁< 0 and h_n> 0 )

When h₁< 0 and h_n > 0, integer solutions have been found to the following 23 types:

( h = -1, 0, 2 )
- [ -12, -10, 1 ] = [ 2, 4, 15 ]
  - By Chen Shuwen in 2017-03-26.
( h = -1, 1, 2 )
- [ -76, 95, 380 ] = [ -126, 210, 315 ]
  - Numerical example, by Ajai Choudhry's parametric solution in 2011. [1101]
  - Parametric solution of multigrade chains of arbitrary length is obtained by Ajai Choudhry in 2011. [1101]
( h = -2, 0, 1 )
- [ -6, -5, 60 ] = [ 4, 15, 30 ]
  - By Chen Shuwen in 2017-03-26.
( h = -2, -1, 1 )
- [ -315, 63, 252 ] = [ -190, 76, 114 ]
  - By applying Chen Shuwen's Theorem 9.
( h = -1, 0, 1, 3 )
- [ -30, 4, 5, 21 ] = [ -28, 3, 10, 15 ]
  - Smallest solution, by Chen Shuwen in 2017-03-26.
( h = -1, 0, 1, 5 )
- [ -891, 85, 286, 702 ] = [ -858, 81, 374, 585 ]
  - By Chen Shuwen in 2017-04-07.
( h = -1, 0, 2, 4 )
- [ -55, -77, 180, 312 ] = [ -40, -132, 143, 315 ]
  - By Chen Shuwen in 2019-02-11.
( h = -1, 1, 2, 3 )
- [ -39, 65, 156, 260 ] = [ -34, 51, 170, 255 ]
  - Numerical example, by Ajai Choudhry's parametric solution in 2011. [1101]
  - Parametric solution of multigrade chains of arbitrary length is obtained by Ajai Choudhry in 2011. [1101]
- [ -3, 4, 21, 28 ] = [ -5, 10, 15, 30 ]
- [ -7, 9, 56, 72 ] = [ -10, 15, 50, 75 ] = [ -12, 21, 44, 77 ] = [ -13, 26, 39, 78 ]
  - The above two sets of solutions are smallest, found by Chen Shuwen in 2019-02-12.
( h = -2, 0, 1, 2 )
- [ -6, 7, 30, 35 ] = [ -5, 14, 15, 42 ]
  - By Chen Shuwen in 2017-03-25.
( h = -2, -1, 0, 2 )
- [ 6, 7, 30, -35 ] = [ 5, 14, 15, -42 ]
  - By Chen Shuwen in 2017-03-25.
( h = -2, -1, 1, 2 )
- [ -230, -92, 23, 46 ] = [ -220, -110, 22, 55 ]
  - By Ajai Choudhry in 2011. [1101]
( h = -2, -1, 1, 3 )
- [ -34775, 2247, 21828, 36594 ] = [ -15246, 2299, 12100, 26741 ]
  - First known solution, by Chen Shuwen in 2023-05-12.
( h = -3, -1, 0, 1 )
- [ -15, 28, 42, 140 ] = [ -14, 20, 84, 105 ]
  - By Chen Shuwen in 2017-03-26.
( h = -3, -1, 1, 2 )
- [ -22464650, 12807900, 28305459, 148976100 ] = [ -9848916, 9359350, 15690675, 152423700 ]
  - First known solution, by Chen Shuwen in 2023-05-12.
( h = -3, -2, -1, 1 )
- [ -340, 204, 85, 51 ] = [ -390, 260, 78, 52 ]
  - By applying Chen Shuwen's Theorem 9.
( h = -4, -2, 0, 1 )
- [ -1155, -2002, 4680, 6552 ] = [ -1144, -2520, 2730, 9009 ]
  - By Chen Shuwen in 2019-02-11.
( h = -5, -1, 0, 1 )
- [ -2210, 2805, 6885, 23166 ] = [ -2295, 3366, 5265, 24310 ]
  - By Chen Shuwen in 2017-04-07.
( h = -1, 0, 1, 2, 4 )
- [ -63, -45, 8, 16, 84 ] = [ -72, -36, 7, 21, 80 ]
  - By Chen Shuwen in 2019-02-05.
( h = -4, -2, -1, 0, 1 )
- [ -112, -80, 60, 315, 630 ] = [ -140, -70, 63, 240, 720 ]
  - By Chen Shuwen in 2019-02-05.
( h = -1, 0, 1, 2, 3, 5 )
- [ -156, -130, 13, 35, 42, 196 ] = [ -147, -140, 14, 26, 52, 195 ]
  - By Chen Shuwen in 2017-03-27.
( h = -1, 1, 2, 3, 4, 5 )
- [ -13, 26, 52, 130, 156, 195 ] = [ -14, 35, 42, 140, 147, 196 ]
  - Numerical example, by Ajai Choudhry's parametric solution in 2011. [1101]
  - Parametric solution of multigrade chains of arbitrary length is obtained by Ajai Choudhry in 2011. [1101]
( h = -5, -3, -2, -1, 0, 1 )
- [ -294, -245, 195, 910, 1092, 2940 ] = [ -273, -260, 196, 735, 1470, 2730 ]
  - By Chen Shuwen in 2017-03-27.
( h = -5, -4, -3, -2, -1, 1 )
- [ -2940, 1470, 735, 294, 245, 196 ] = [ -2730, 1092, 910, 273, 260, 195 ]
  - By applying Chen Shuwen's Theorem 9.

Parametric solutions

Parametric solutions have been found to several types, includes the following:

( h = -1, 1, 2, 3 ) and ( h = -1, 0, 1, 3 )
- Ajai Choudhry found the below parametric solution in 2011 for ( h = -1, 1, 2, 3 ). [1101]
  - [ a1, a2, a3, a4 ] = [ b1, b2, b3, b4 ] ( h = -1, 1, 2, 3 )
    - a1 = m(m-n)(p^2+q^2)
    - a2 = n(m+n)(p^2+q^2)
    - a3 = (n-m)n(p^2+q^2)
    - a4 = m(m+n)(p^2+q^2)
    - b1 = p(p-q)(m^2+n^2)
    - b2 = q(p+q)(m^2+n^2)
    - b3 = (q-p)q(m^2+n^2)
    - b4 = p(p+q)(m^2+n^2)
- Chen Shuwen found in 2017-03-26 that the parametric solution for (h = -1,1,2,3) can lead to parametric solution for (h = -1,0,1,3).
  - [ a1, a2, -b3, -b4 ] = [ b1, b2, -a3, -a4 ] ( h = -1, 0, 1, 3 )
  - [ a1, a2, -b1, -b2 ] = [ b3, b4, -a3, -a4 ] ( h = -1, 0, 1, 3 )
( h = -1, 1, 2, 3, 4, 5 ) and ( h = -1, 0, 1, 2, 3, 5 )
- Ajai Choudhry found the below parametric solution in 2011 for ( h = -1, 1, 2, 3, 4, 5 ). [1101]
  - [ a1, a2, a3, a4, a5, a6 ] = [ b1, b2, b3, b4, b5, b6 ] ( h = -1, 1, 2, 3, 4, 5 )
    - a1 = m(m-n)(p^2+pq+q^2)
    - a2 = n(m+2n)(p^2+pq+q^2)
    - a3 = (m+n)(2m+n)(p^2+pq+q^2)
    - a4 = n(n-m)(p^2+pq+q^2)
    - a5 = m(2m+n)(p^2+pq+q^2)
    - a6 = (m+n)(m+2n)(p^2+pq+q^2)
    - b1 = p(p-q)(m^2+mn+n^2)
    - b2 = q(p+2q)(m^2+mn+n^2)
    - b3 = (p+q)(2p+q)(m^2+mn+n^2)
    - b4 = q(q-p)(m^2+mn+n^2)
    - b5 = p(2p+q)(m^2+mn+n^2)
    - b6 = (p+q)(p+2q)(m^2+mn+n^2)
- Chen Shuwen found in 2017-03-27 that the parametric solution for (h = -1,1,2,3,4,5) can lead to parametric solution for (h = -1,0,1,2,3,5).
  - [ a1, a2, a3, -b4, -b5, -b6 ] = [ b1, b2, b3, -a4, -a5, -a6 ] ( h = -1, 0, 1, 2, 3, 5 )
  - [ a1, a2, a3, -b1, -b2, -b3 ] = [ b4, b5, b6, -a4, -a5, -a6 ] ( h = -1, 0, 1, 2, 3, 5 )