Results and discussion on the k<0 cases

`Equal Sums of Like Powers`

Ideal positive integer solutions of k < 0 case

a₁^k + a₂^k + ... + a_n₊₁^k = b₁^k + b₂^k + ... + b_n₊₁^k ( k = k₁ , k₂, ... , k_n with k₁ <0)

Introduction

General theorems

Ideal positive integer solutions ( all k < 0 )

Ideal positive integer solutions ( kn = 0 and all others k < 0 )

Ideal positive integer solutions ( k1 < 0 and kn > 0 )

Introduction

Guo Xianqiang studied the k < 0 cases of the Equal sums of like powers system first. He gave some numerial examples such as the following solution (not an ideal one, as each side has seven numbers) for ( k = 0, -1, -2, -3, -4 ) before March of 2001:

[684, 855, 1140, 2160, 4104, 4560, 20520 ] = [ 720, 760, 1368, 2052, 2565, 10260, 13680 ]

Note: Here k = 0 is used to denote the equal products equation. Please refer to Equal products and equal sums of like powers for details.

In 2001, on Guo Xianqiang's Website on Maths ( in Chinese ), he ask such an interesting question:

If k₁< 0 and k_n> 0 , is there solution in positive integer? As an answer, Chen Shuwen solved ( k = -1, 1 ) in May of 2001.

In March of 2001, Chen Shuwen noticed that ( Theorem 9 ), if all the k<=0, ( k = k₁, k₂ , ... , k_n ), solutions can be easily obtained by simply transforming from a corresponding known type:

[ a₁ , a₂ , ... , a_m ] = [ b₁ ,b₂ , ... , b_m ] ( k = k₁, k₂ , ... , k_n )

<=> [ C/a₁ , C/a₂ , ... , C/a_m ] = [ C/b₁ ,C/b₂ , ... , C/b_m ] ( k = -k₁, -k₂ , ... , -k_n )

where C is the least common multiple of all { a_i , b_i }

For example, from the following solution of ( k = 1, 2, 4, 6 ):

[ 3, 7, 10, 16, 16 ] = [ 4, 5, 12, 14, 17 ]

We can get a solution of ( k = -1, -2, -4, -6 ) without any difficult ( here C=28560 ):

[ 1680, 2040, 2380, 5712, 7140 ] = [ 1785, 1785, 2856, 4080, 9520 ]

The above transformation is also true for the case of k₁< 0 and k_n> 0. For example:

[ a₁, a₂ , a₃ ] = [ b₁, b₂, b₃ ] ( k = -1, 2 )

<=> [ C/a₁ , C/a₂ , C/a₃ ] = [ C/b₁ ,C/b₂ , C/b₃ ] ( k = -2, 1 )

In this page, we will only consider the following system of k₁< 0 case and the ideal positive integer solutions ( m = n + 1 ):

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

General theorems

Theorem 9 ( By Chen Shuwen, March 2001 )

[ a₁ , a₂ , ... , a_m ] = [ b₁ ,b₂ , ... , b_m ] ( a_i <>0 , b_i<>0 )

( k = k₁, k₂ , ... , k_n )

then

[ C/a₁ , C/a₂ , ... , C/a_m ] = [ C/b₁ ,C/b₂ , ... , C/b_m ]

( k = -k₁, -k₂ , ... , -k_n )

where C is the least common multiple of all { a_i , b_i }.

Ideal positive integer solutions ( all k < 0 )

When all k < 0, ideal positive integer solutions have been found to the following 42 types (by Chen Shuwen, using Theorem 9):

( k = -1 )

[ 3, 12 ] = [ 4, 6 ]

( k = -2 )

[ 10, 55 ] = [ 11, 22 ]

( k = -3 )

[ 15, 180 ] = [ 18, 20 ]

( k = -4 )

[ 525749, 1407938 ] = [ 619913, 624574 ]

( k = -2, -1 )

[ 21, 42, 56 ] = [ 24, 28, 84 ]

( k = -3, -1 )

[ 120, 195, 520 ] = [ 130, 156, 780 ]

( k = -4, -1 )

[ 13300, 25935, 74100 ] = [ 13650, 20748, 172900 ]

( k = -5, -1 )

[ 21216, 31824, 77792 ] = [ 21879, 27456, 107712 ]

( k = -3, -2 )

[ 6210, 10074, 25185 ] = [ 6570, 8395, 453330 ]

( k = -4, -2 )

[ 90, 165, 198 ] = [ 99, 110, 990]

( k = -6, -2 )

[ 6270, 9614, 14421 ] = [ 6555, 7590, 48070 ]

( k = -3, -2, -1 )

[ 120, 132, 440, 660 ] = [ 110, 165, 264, 1320 ]

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

[ 2145, 2310, 6006, 10010 ] = [ 2002, 2730, 4290, 15015 ]

( k = -5, -2, -1 )

[ 1076712, 1118936, 4076124, 7133217 ] = [ 983892, 1463224, 2038062, 57065736 ]

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

[ 75600, 81200, 156600, 274050 ] = [ 73080, 87696, 137025, 313200 ]

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

[ 188678700, 224349300, 835077950, 901884186 ] = [ 178945275, 302645700, 322101495, 15031403100 ]

( k = -5, -3, -1 )

[ 15249, 15249, 35581, 106743 ] = [ 13923, 18837, 24633, 320229 ]

( k = -7, -3, -1 )

[ 872401691935000, 939402141875608, 1663247418335000, 2552723211618500 ] = [ 841155212997500, 1055982623511250, 1325341622285000, 3190904014523125 ]

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

[ 11041656066882045, 12344254963416090, 33421609706920578, 46265451453795015 ] = [ 10582456759752735, 13737716873396802, 23037529324729205, 260377657019032410 ]

( k = -6, -4, -2 )

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

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

[ 684, 855, 1140, 3420, 4560 ] = [ 720, 760, 1368, 2280, 6840 ]

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

[ 11934，15912，19656，83538，111384 ] = [ 12376，13923，25704，41769，334152 ]

( k = -6, -3, -2, -1 )

[ 43752555, 56947770, 58814910, 239180634, 398634390 ] = [ 44292710, 51995790, 65231082, 199317195, 512529930 ]

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

[ 1680, 2040, 2380, 5712, 7140 ] = [ 1785, 1785, 2856, 4080, 9520 ]

( k = -7, -5, -3, -1 )

[ 245126961, 299599619, 313534485, 1225634805, 1497998095 ] = [ 254377035, 264352605, 364377915, 709578045, 4493994285 ]

( k = -8, -6, -4, -2 )

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

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

[ 13104, 13650, 20475, 23400, 54600, 65520 ] = [ 12600, 15600, 16380, 32760, 36400, 81900 ]

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

[ 41491979760, 42111263040, 64123968720, 91014665280, 188096974912, 201532473120 ] = [ 40306494624, 44085228495, 60030949440, 117560609320, 122671940160, 256495874880 ]

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

[ 31270088385360, 31329992769240, 44683760179080, 52926395551920, 79389593327880, 181713958061592 ] = [ 30285659676932, 32972290777305, 41298626832180, 61947940248270, 70189940882160, 187979956615440 ]

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

[ 1081404, 1113210, 1401820, 1992060, 9462285, 12616380 ] = [1051365, 1220940, 1261638, 2226420, 5407020, 37849140 ]

( k = -9, -7, -5, -3, -1 )

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

( k = -10, -8, -6, -4, -2 )

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

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

[ 187849200, 225419040, 250465600, 405754272, 461084400, 1067774400, 2898244800 ] = [ 189604800, 213554880, 289824480, 316995525, 614779200, 845321400, 3381285600 ]

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

[ 28201256396610842831816398063, 30833029637648634203129547745, 46355572756464865452573033955, 66733666126633578582219001357, 88919528743931285445964632415, 347427849422164507051758718405, 426588625239872875747096147915 ] = [ 28343567194238820171590072065, 30498191306742042700471127317, 50525489346251809871095345855, 55519771653953800962142661755, 114238987776101549776341002323, 199411250851774894580003524765, 783732590556975748465595248495 ]

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

[ 1297752271862287613603, 1563107368984288914787, 1573159506405409743821, 2534987598404572177867, 3240083486702532651181, 15782342144905884849301, 16870779534209738976839 ] = [ 1311669186305851019647, 1434758376809625895391, 1933804768743408815527, 1949213571681603308081, 5376402269143762970641, 5497220297663847531779, 69893229498868918618333 ]

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

[ 14564172, 14861400, 18205215, 23490600, 34676600, 60684050, 242736200, 364104300 ] = [ 14278600, 15493800, 17338300, 26007450, 30342025, 72820860, 145641720, 728208600 ]

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

[ 4103980139754883042177329807, 4189301972182219487960642277, 5050261274735958831351049963, 5551113632561012599749501199, 8502338601770664867970755001, 9460348585068767951685769649, 14708425172406186669409262301, 25507015805311994603912265003 ] = [ 4022064368502290566285566737, 4507951339193842446776440571, 4548655188757669466611893759, 6277427565793294622146632197, 7120333033991687539608017439, 11923397920826317004195674173, 12673297161129858954145087643, 26169535696359059398819077081 ]

( k = -8, -7, -6, -5, -4, -3, -2, -1 )

[ 6297380984594700, 7161021805339116, 7415259265883700, 10803265654606425, 11090078017118100, 18987557817187050, 29837590855579650, 69621045329685850, 626589407967172650 ] = [ 6329185939062350, 6885597889749150, 7931511493255350, 9352080715927950, 14404354206141900, 14918795427789825, 40425123094656300, 50127152637373812, 1253178815934345300 ]

( k = -9, -8, -7, -6, -5, -4, -3, -2, -1 )

[ 540067769362905381082485, 540937443709705550778270, 680004357375965884682805, 754881241671296959625406, 872525071542148433852742, 1382395689480358629766690, 1835640177834574573952490, 2506881735400948858457505, 47988878934818163861900810, 55987025423954524505550945 ] = [ 535761008841670090962210, 546214882184922190298058, 669167634549257265006585, 811406165564558326167405, 813370829403697692574590, 1562428616482451846666538, 1569729684783771715108905, 2666048829712120214550045, 25840165580286703617946590, 335922152543727147033305670 ]

( k = -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1 )

[ 29207186880174600297, 29403208268632147950, 32095809758433626700, 35618520585578780850, 44030934995238090900, 46856449540387059300, 74890222769678462300, 83449105371927429420, 151071656276765173950, 282650195614592906100, 1460359344008730014850, 2190539016013095022275 ] = [ 28918006812054059700, 30007383781001301675, 31405577290510322900, 36815781781732689450, 41136882929823380700, 50357218758921724650, 67401200492710616070, 96287429275300880100, 132759940364430001350, 350486242562095203564, 730179672004365007425, 8762156064052380089100 ]

Ideal positive integer solutions ( k_n= 0 and all others k< 0 )

When k_n= 0 and all others k< 0, ideal positive integer solutions have been found to the following 23 types (by Chen Shuwen, using Theorem 9):

( k = -1, 0 )

[ 3, 10, 20 ] = [ 4, 5, 30 ]

( k = -2, 0 )

[ 28, 70, 80 ] = [ 35, 40, 112 ]

( k = -3, 0 )

[ 15, 216, 240 ] = [ 18, 20, 2160 ]

( k = -4, 0 )

[ 957, 1798, 4092 ] = [ 1023, 1276, 5394 ]

( k = -2, -1, 0 )

[ 21, 42, 60, 280 ] = [ 24, 28, 105, 210 ]

( k = -3, -1, 0 )

[ 120, 132, 330, 440 ] = [ 110, 165, 240, 528 ]

( k = -3, -2, 0 )

[ 3544146, 5563485, 13345695, 67711842 ] = [ 3639735, 5036418, 20399445, 47649074 ]
- By Chen Shuwen in 2022-10-13.

( k = -4, -1, 0 )

[ 60, 126, 140, 1008 ] = [ 63, 84, 280, 720 ]

( k = -4, -2, 0 )

[ 69483, 81673, 263511, 607221 ] = [ 67469, 87837, 208449, 735057 ]

( k = -5, -1, 0 )
- [ 6006, 10230, 14105, 71610 ] = [ 6510, 7161, 30030, 44330 ]
( k = -6, -2, 0 )
- [ 11712265, 15189721, 28371045, 142587381 ] = [ 11991885, 13979155, 35936657, 119465103 ]
  - By Chen Shuwen in 2017-05-19.
( k = -3, -2, -1, 0 )

[ 850, 1300, 1768, 5525, 8840 ] = [ 884, 1105, 2600, 3400, 11050 ]

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

[ 54, 84, 72, 189, 168 ] = [ 63, 56, 108, 126, 216 ]
[ 30, 44, 44, 165, 165 ] = [ 33, 33, 60, 110, 220 ]

( k = -6, -2, -1, 0 )

[ 36234939, 39770055, 46754760, 59025240, 60672456 ] = [ 37164040, 37920285, 51155208, 52599105, 63632088 ]
- By Chen Shuwen in 2017-05-31.

( k = -6, -4, -2, 0 )

[ 144, 180, 225, 800, 2400 ] = [ 150, 160, 288, 450, 3600 ]
- By Chen Shuwen in 2017.

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

[ 183128400, 193302200, 194201280, 217464975, 238590144, 247060800 ] = [ 185570112, 186398550, 200736900, 214119360, 242751600, 244171200 ]

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

[ 330600, 404985, 465500, 809970, 1421000, 2892750 ] = [ 349125, 355250, 578550, 647976, 1653000, 2699900 ]
- By Chen Shuwen, in 2017-04-17.

( k = -6, -4, -2, -1, 0 )

[ 480, 576, 704, 1320, 1980, 3960 ] = [ 495, 528, 880, 960, 2880, 3168 ]
[ 720, 920, 1035, 2208, 2760, 6624 ] = [ 736, 828, 1380, 1440, 4140, 5520 ]

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

[ 50483203485, 52111693920, 53848750384, 58693274640, 59726606940, 68534773216, 70529548320 ] = [ 50938007120, 51168495840, 55072585620, 57016323936, 60796976240, 68258979360, 70676484879 ]

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

[ 324720, 364320, 373428, 466785, 663872, 829840, 905280 ] = [ 331936, 339480, 414920, 432960, 728640, 746856, 933570 ]
- By Chen Shuwen, in 2017-05-26.

( k = -8, -6, -4, -2, -1, 0 )

[ 1270470709119612, 1415077131295828, 1514981029346694, 2077573454037774, 2140014186262953, 3151892121417468, 3584646188431308 ] = [ 1286116900118622, 1351587907113444, 1697172680763252, 1736685570226698, 2522528799266476, 2858559095519127, 3703286960625252 ]
- By Chen Shuwen, in 2023-02-12.

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

[ 36772168119182031123, 37714627576058989941, 39025241062927066409, 39458253446426950641, 41898970154453359959, 42652774714953270483, 44475352894539600957, 44708330122902825687 ] = [ 36740508912880539921, 38084873808398703363, 38255581445370459093, 40315636374407885799, 41052526312949251677, 43611753809208734919, 43701076397655655011, 44943761034555815817 ]

( k = -7, -6, -5, -4, -3, -2, -1, 0 )

[ 8769115037417904, 9081765998004960, 9335358067757280, 9979776441837540, 10828259459159760, 11966801281487520, 12858558107752215, 13786495290785880, 13822119309625120 ] = [ 8797960810567305, 8975100961115640, 9569159522048160, 9743461152686560, 11029196232628704, 11730614414089740, 13273350304776480, 13306368589116720, 14003037101635920 ]
- By Chen Shuwen, in 2023-04-08.

Ideal positive integer solutions ( k₁< 0 and k_n> 0 )

When k₁ < 0 and k_n > 0, ideal positive integer solutions have been found to the following 33 types:

( k = -1, 1 )

[ 4, 10, 12 ] = [ 5, 6, 15 ]

First known solution, smallest solution, by Chen Shuwen in 2001-05-06.

[ 6, 14, 14 ] = [ 7, 9, 18 ]

By Chen Shuwen in 2001.

[ 3, 40 ] = [ 4, 15, 24 ] = [ 5, 8, 30 ]

By Chen Shuwen in 2001.

( k = -1, 2 )

[ 35, 65, 84 ] = [ 39, 52, 91 ]

First known solution, smallest solution, by Chen Shuwen in 2001-05-08.

[ 26, 143, 165 ] = [ 35, 55, 210 ]

By Chen Shuwen in 2001.

( k = -2, 1 )

[ 60, 105, 140 ] = [ 65, 84, 156 ]

First known solution, smallest solution, by Chen Shuwen in 2001-05-08.

[ 143, 546, 858 ] = [ 182, 210, 1155 ]

By Chen Shuwen in 2001.

( k = -2, 2 )

[ 77, 1057, 1661 ] = [ 91, 143, 1963 ]

Numerical example, by Ajai Choudhry's parametric solution in 2011. [1101]

[ 22, 220, 275 ] = [ 28, 35, 350 ]
- Smallest solution, by Chen Shuwen in 2017-03-26.

( k = -1, 0, 2 )

[ 6, 6, 52, 65 ] = [ 4, 15, 26, 78 ]

There is only 1 solutions in the range of max{a_i,b_i}<=130, by Chen Shuwen in 2017.

( k = -1, 0, 1 )

[ 4, 10, 18, 45 ] = [ 5, 6, 30, 36 ]

First known solution, by Guo Xianqiang in 2001-05-08.

[ 5, 15, 21, 63 ] = [ 7, 7, 45, 45 ]

By Chen Shuwen in 2001.

( k = -1, 1, 2 )

[ 15, 28, 48, 70 ] = [ 16, 24, 55, 66 ]
[ 34, 42, 63, 72 ] = [ 35, 40, 68, 68 ]
- There are 7 solutions in the range of max{a_i,b_i}<=110, by Chen Shuwen in 2017.

( k = -1, 1, 3 )

[ 4, 5, 21, 28 ] = [ 3, 10, 15, 30 ]
[ 7, 15, 78, 91 ] = [ 6, 25, 60, 100 ]
- There are 14 solutions in the range of max{a_i,b_i}<=100, by Chen Shuwen in 2017.
- Solution of ( h = -1, 1, 2, 3 ) may lead to ( k = -1, 1, 3 ).

( k = -1, 1, 5 )
- [ 85, 286, 702, 858 ] = [ 81, 374, 585, 891 ]
  - Smallest solution, by Chen Shuwen in 2017-04-07.
( k = -2, 0, 1 )

[ 10, 30, 52, 195 ] = [ 12, 15, 130, 130 ]

Based on the solution of ( k = -1, 0, 2 ), by Chen Shuwen in 2017.

( k = -2, 0, 2 )

[ 3553, 15521, 26125, 114125 ] = [ 4565, 5225, 77605, 88825 ]

First known solution, by Chen Shuwen in 2014-09-06.

[ 6, 7, 30, 35 ] = [ 5, 14, 15, 42 ]

Smallest solution, by Chen Shuwen in 2017-03-25.

( k = -2, -1, 1 )

[ 126, 168, 342, 504 ] = [ 133, 152, 399, 456 ]
[ 182, 273, 525, 910 ] = [ 175, 350, 390, 975 ]
- Based on the solution of ( k = -1, 1, 2 ), by Chen Shuwen in 2017.

( k = -3, 0, 3 )

[ 16, 20, 108, 135 ] = [ 15, 24, 90, 144 ]
- First known solution, by Chen Shuwen in 2017-03-25.
[ 231, 693, 750, 2250 ] = [ 275, 297, 1750, 1890 ]
- By Chen Shuwen in 2023-05-02.

( k = -3, -1, 1 )

[ 14, 28, 42, 140 ] = [ 15, 20, 84, 105 ]
[ 25, 50, 75, 195 ] = [ 26, 39, 130, 150 ]
- Based on the solution of ( k = -1, 1, 3 ), by Chen Shuwen in 2017.

( k = -5, -1, 1 )

[ 2210, 3366, 5265, 24310 ] = [ 2295, 2805, 6885, 23166 ]
- Based on the solution of ( k = -1, 1, 5 ), by Chen Shuwen in 2017-04-07.

( k = -1, 0, 1, 2 )
- [ 8, 8, 16, 28, 35 ] = [ 7, 10, 14, 32, 32 ]
- [ 5, 5, 14, 28, 40 ] = [ 4, 8, 10, 35, 35 ]
  - By Chen Shuwen in 2017.
( k = -1, 0, 1, 3 )
- [ 6, 7, 28, 50, 75 ] = [ 5, 10, 21, 60, 70 ]
- [ 14, 18, 52, 63, 91 ] = [ 13, 21, 42, 78, 84 ]
  - By Chen Shuwen in 2017.
( k = -1, 1, 2, 3 )
- [ 266, 494, 494, 1463, 1547 ] = [ 287, 374, 611, 1394, 1598 ]
  - First know solution, by Chen Shuwen in 2022-09-22.
( k = -2, 0, 2, 4 )
- [ 135, 240, 612, 731, 1376 ] = [ 129, 340, 387, 864, 1360 ]
  - First known solution, by Chen Shuwen in 2017-05-31.
( k = -2, -1, 0, 1 )
- [ 7, 10, 20, 56, 56 ] = [ 8, 8, 28, 35, 70 ]
- [ 32, 40, 70, 140, 140 ] = [ 35, 35, 80, 112, 160 ]
  - Based on the solution of ( k = -1, 0, 1, 2 ), by Chen Shuwen in 2017.
( k = -2, -1, 1, 2 )
- [ 25, 35, 75, 100, 175 ] = [ 24, 42, 56, 120, 168 ]
  - First known solution, smallest solution, by Chen Shuwen in 2017-03-30.
- [ 212, 318, 477, 848, 1166 ] = [ 216, 297, 528, 792, 1188 ]
  - Second known solution, by Chen Shuwen in 2019-12-18.
- [ 3300, 4675, 6732, 11968, 15048 ] = [ 3400, 4275, 7600, 10944, 15504 ]
  - Third known solution, by Chen Shuwen in 2022-10-10.
( k = -3, -1, 0, 1 )
- [ 36, 52, 63, 182, 234 ] = [ 39, 42, 78, 156, 252 ]
- [ 28, 42, 75, 300, 350 ] = [ 30, 35, 100, 210, 420 ]
  - Based on the solution of ( k = -1, 0, 1, 3 ), by Chen Shuwen in 2017.
( k = -3, -2, -1, 1 )
- [ 779779, 893893, 2039422, 3331783, 4341766 ] = [ 805486, 851734, 2522443, 2522443, 4684537 ]
  - By Chen Shuwen in 2022.
( k = -4, -2, 0, 2 )
- [ 2295, 4320, 5160, 13158, 23392 ] = [ 2322, 3655, 8160, 9288, 24480 ]
  - By Chen Shuwen in 2017-05-31.
( k = -1, 0, 1, 2, 3 )
- [ 50, 55, 70, 88, 91, 104 ] = [ 52, 52, 77, 77, 100, 100 ]
  - By Chen Shuwen in 2017.
( k = -1, 0, 1, 2, 4 )
- [ 10, 14, 24, 65, 117, 132 ] = [ 11, 12, 26, 63, 120, 130 ]
  - First known solution, smallest solution, by Chen Shuwen in 2017-05-26.
- [ 36, 51, 72, 130, 195, 238 ] = [ 40, 42, 85, 117, 204, 234 ]
  - Second known solution, by Chen Shuwen in 2022-11-09.
( k = -2, -1, 0, 1, 2 )
- [ 14, 18, 18, 35, 35, 45 ] = [ 15, 15, 21, 30, 42, 42 ]
- [ 6, 11, 11, 42, 42, 77 ] = [ 7, 7, 21, 22, 66, 66 ]
  - By Chen Shuwen in 2017-03-29.
( k = -3, -1, 0, 1, 3 )
- [ 8, 12, 14, 36, 42, 63 ] = [ 9, 9, 21, 24, 56, 56 ]
- [ 7, 11, 14, 44, 56, 88 ] = [ 8, 8, 22, 28, 77, 77 ]
  - By Chen Shuwen in 2017-03-30.
( k = -3, -2, -1, 0, 1 )
- [ 1925, 2200, 2275, 2860, 3640, 4004 ] = [ 2002, 2002, 2600, 2600, 3850, 3850 ]
  - Based on the solution of ( k = -1, 0, 1, 2, 3 ), by Chen Shuwen in 2017.
( k = -4, -2, -1, 0, 1 )
- [ 2730, 3080, 5544, 15015, 25740, 36036 ] = [ 2772, 3003, 5720, 13860, 30030, 32760 ]
  - By Chen Shuwen in 2017-05-26.
( k = -1, 0, 1, 2, 3, 4 )
- [ 9, 17, 21, 51, 99, 143, 143 ] = [ 11, 11, 33, 39, 117, 119, 153 ]
  - First known solution, smallest solution, by Chen Shuwen in 2017-05-21.
- [ 56, 77, 99, 152, 174, 228, 261 ] = [ 57, 72, 116, 126, 203, 209, 264 ]
  - Second known solution, by Chen Shuwen in 2017.
( k = -4, -3, -2, -1, 0, 1 )
- [ 1001, 1287, 1309, 3927, 4641, 13923, 13923 ] = [ 1071, 1071, 1547, 3003, 7293, 9009, 17017 ]
  - Based on the solution of ( k = -1, 0, 1, 2, 3, 4 ), by Chen Shuwen in 2017-05-21.
( k = -3, -2, -1, 0, 1, 2, 3 )
- [ 22, 22, 33, 33, 56, 56, 84, 84 ] = [ 21, 24, 28, 42, 44, 66, 77, 88 ]
  - First known solution, smallest solution, by Chen Shuwen in 2017-03-30.
- [ 7980, 8060, 8151, 8360, 8463, 8680, 8778, 8866 ] = [ 8008, 8008, 8246, 8246, 8580, 8580, 8835, 8835 ]
  - Second known solution, by Chen Shuwen in 2019.