$abc$-triples of high merit
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Numbers
$m$
value
38.665728322551
$a$:
1836663805028956328780786321829823
38.665728322551
$b$:
341369987831125687798753779436199529840279552
38.665728322551
$c$:
341369987832962351603782735764980316162109375
37.947548141246
$a$:
785717171553879
37.947548141246
$b$:
320913062252451680627140051026068480
37.947548141246
$c$:
320913062252451680627925768197622359
36.991189057610
$a$:
110694767111422864255245173136013049441
36.991189057610
$b$:
5189591376653295184950594460805637341184
36.991189057610
$c$:
5300286143764718049205839633941650390625
36.833494984114
$a$:
1047285008500551591948178915459225
36.833494984114
$b$:
146839682795137930866165342911043074963960437671
36.833494984114
$c$:
146839682795138978151173843462635023142875896896
35.323779187323
$a$:
333101198767152070635432044699
35.323779187323
$b$:
4716418511051923209358717412852717539900164721868800000
35.323779187323
$c$:
4716418511051923209358717745953916307052235357300844699
35.155815204829
$a$:
8654525279998779296875
35.155815204829
$b$:
229727166528260169448321941
35.155815204829
$c$:
229735821053540168227618816
35.120565929294
$a$:
464743623105438551
35.120565929294
$b$:
6045120559074418176746593449
35.120565929294
$c$:
6045120559539161799852032000
34.760005211471
$a$:
947879669685667223
34.760005211471
$b$:
8873939344779791540547225129034332777
34.760005211471
$c$:
8873939344779791541495104798720000000
34.402819297495
$a$:
146767394485224241
34.402819297495
$b$:
13669290314405085785446416384
34.402819297495
$c$:
13669290314551853179931640625
33.897569791317
$a$:
393927551841835417600000
33.897569791317
$b$:
10704872737453891276395489673
33.897569791317
$c$:
10705266665005733111813089673
33.147062346241
$a$:
223270703038574259899031
33.147062346241
$b$:
649307365977022004946459915625
33.147062346241
$c$:
649307589247725043520719814656
32.626809236802
$a$:
19954407093901006131733888
32.626809236802
$b$:
1475498869662093155080293491744105475
32.626809236802
$c$:
1475498869682047562174194497875839363
32.590930575953
$a$:
7958374958079
32.590930575953
$b$:
698042392297968606058159789361760359205584406924356012673597440
32.590930575953
$c$:
698042392297968606058159789361760359205584406924363971048555519
32.227570248394
$a$:
134783940126171875
32.227570248394
$b$:
92834181056683026002118828883968
32.227570248394
$c$:
92834181056683160786058955055843
31.883256109898
$a$:
2707160810382798173125
31.883256109898
$b$:
362056449426413786635575296
31.883256109898
$c$:
362059156587224169433748421
31.820743703551
$a$:
553611389360227869798332916832
31.820743703551
$b$:
109098170963255825195876997895538806915283203125
31.820743703551
$c$:
109098170963255825749488387255766676713616119957
31.543672018743
$a$:
2477678547239
31.543672018743
$b$:
613474843408551921511
31.543672018743
$c$:
613474845886230468750
31.476415500151
$a$:
55204820542052128964537745408
31.476415500151
$b$:
183636889121597576688232421875
31.476415500151
$c$:
238841709663649705652770167283
31.369024029242
$a$:
916042593694120132608
31.369024029242
$b$:
179239666784997133807373046875
31.369024029242
$c$:
179239667701039727501493179483
31.322762756756
$a$:
98510087171258464329115223138227
31.322762756756
$b$:
14373412790223329808678441598877578664935615190196783648
31.322762756756
$c$:
14373412790223329808678540108964749923399944305419921875
30.195600519809
$a$:
145114547144850771075432984934127871829439
30.195600519809
$b$:
478370551231697723988140468339630762475874048170561
30.195600519809
$c$:
478370551376812271132991239415063747410001920000000
30.060367969121
$a$:
25220932830213426279247196812890625
30.060367969121
$b$:
17485613666400818352222466948402205184
30.060367969121
$c$:
17510834599231031778501714145215095809
30.031620220480
$a$:
922815434786010581540958753029247178966000261
30.031620220480
$b$:
867073024123444714825006372969734121512218816498870046793708668452864
30.031620220480
$c$:
867073024123444714825007295785168907522800357457623076040887634453125
30.012089370634
$a$:
20443728973364345302466693353
30.012089370634
$b$:
3891320061619085979788813136822272
30.012089370634
$c$:
3891340505348059344134115603515625
29.900747650546
$a$:
8149759554184307428900880126953125
29.900747650546
$b$:
184801588605534844978995877224906752
29.900747650546
$c$:
192951348159719152407896757351859877
29.823682424800
$a$:
44790692380548068359375
29.823682424800
$b$:
3417300183328464869570036529
29.823682424800
$c$:
3417344974020845417638395904
29.586609728186
$a$:
3971503294809916729784891216754913330078125
29.586609728186
$b$:
199229281157617277256340351246053758806673852944
29.586609728186
$c$:
199233252660912087173070136137270513720003931069
29.499354472809
$a$:
9168840339054545809
29.499354472809
$b$:
5785662043817054242167132959765125700287109375
29.499354472809
$c$:
5785662043817054242167132968933966039341655184
29.255676256538
$a$:
8628951430756189421875
29.255676256538
$b$:
543059049820697074202106338131661
29.255676256538
$c$:
543059049829326025632862527553536
29.136264619492
$a$:
1871533462882238491
29.136264619492
$b$:
192664186774168804232699779530337812480
29.136264619492
$c$:
192664186774168804234571312993220050971
28.758159224396
$a$:
41384332596026661258466773453425380333980777
28.758159224396
$b$:
48326710094123879889444068603515625000000000
28.758159224396
$c$:
89711042690150541147910842056941005333980777
28.670515071014
$a$:
11745463940603527909
28.670515071014
$b$:
1092734360998619466244472091
28.670515071014
$c$:
1092734372744083406848000000
28.642751010861
$a$:
2704798098231340213727
28.642751010861
$b$:
2282915619421863094395648
28.642751010861
$c$:
2285620417520094434609375
28.590541976968
$a$:
522126500272673131
28.590541976968
$b$:
74392346486376275746647573022869
28.590541976968
$c$:
74392346486376797873147845696000
28.425448061483
$a$:
1687985790084250677387
28.425448061483
$b$:
141949861450245248022563279756750812935501119488
28.425448061483
$c$:
141949861450245248022563281444736603019751796875
28.381327711112
$a$:
457311092558307719056799
28.381327711112
$b$:
13907329461548919677734375000
28.381327711112
$c$:
13907786772641477985453431799
28.370793093797
$a$:
5928730440118282309
28.370793093797
$b$:
8026769053317573738092929801519104
28.370793093797
$c$:
8026769053317579666823369919801413
28.360507672661
$a$:
1510172634014460572671018731233386121
28.360507672661
$b$:
38110972058470684008789302354161863817462824338932629504
28.360507672661
$c$:
38110972058470684010299474988176324390133843070166015625
28.315779017257
$a$:
130688410876456206336
28.315779017257
$b$:
18060557607961536446730761769935514491059289
28.315779017257
$c$:
18060557607961536446730892458346390947265625
28.288618742727
$a$:
2203646221741844640625
28.288618742727
$b$:
71649217146644876178683022088368
28.288618742727
$c$:
71649217148848522400424866728993
28.137228932816
$a$:
2831054986033663935641
28.137228932816
$b$:
3035095189189615974361751964815916131618664375
28.137228932816
$c$:
3035095189189615974361754795870902165282600016
27.991897989200
$a$:
4947958397180247015424
27.991897989200
$b$:
81281131236689787773584530930848687701
27.991897989200
$c$:
81281131236689792721542928111095703125
27.936739811012
$a$:
259217962079863808
27.936739811012
$b$:
13821594916284084320068359375
27.936739811012
$c$:
13821594916543302282148223183
27.925016401010
$a$:
16549979536155372440576842641546988596745200315567
27.925016401010
$b$:
23951364622681064001554421864319328979187184666200812488706743607692055543808
27.925016401010
$c$:
23951364622681064001554421880869308515342557106777655130253732204437255859375
27.922330220105
$a$:
25161368403091
27.922330220105
$b$:
570243460709396737065659
27.922330220105
$c$:
570243460734558105468750
27.903194438187
$a$:
563724163211882114410400390625
27.903194438187
$b$:
2834612346924429283958528844636703
27.903194438187
$c$:
2835176071087641166072939245027328
27.826281395126
$a$:
2395269484225019653222026969088
27.826281395126
$b$:
2003563160751184488307461284060069849
27.826281395126
$c$:
2003565556020668713327114506087038937
27.767685453595
$a$:
121588887369723164152821276243876547801631
27.767685453595
$b$:
398703807810572411498315063055075847178723756123452198369
27.767685453595
$c$:
398703807810572533087202432778240000000000000000000000000
27.648185902902
$a$:
8168354035667660710457
27.648185902902
$b$:
201889741674926757812500000
27.648185902902
$c$:
201897910028962425473210457
27.455247272893
$a$:
4356977468
27.455247272893
$b$:
24329989780515009588423793281389467201263427734375
27.455247272893
$c$:
24329989780515009588423793281389467201267784711843
27.376599872651
$a$:
1268199764913417288439021568
27.376599872651
$b$:
3097532453432709681191796723
27.376599872651
$c$:
4365732218346126969630818291
27.300573135561
$a$:
32790610852822302675772069469399953243440854586259736430583993109870250793463
27.300573135561
$b$:
1505981050682367209644189703455091096976881693629334712531714210123638916015625
27.300573135561
$c$:
1538771661535189512319961772924491050220322548215594448962298203233509166809088
27.241394896864
$a$:
23819819904139264
27.241394896864
$b$:
45729959189003560490241156111
27.241394896864
$c$:
45729959189027380310145295375
27.135587155137
$a$:
695606563606442148006101677581923
27.135587155137
$b$:
57576591665034362126590541368210176398589952
27.135587155137
$c$:
57576591665729968690196983516216278076171875
27.128209540870
$a$:
2063261961823521511058423651
27.128209540870
$b$:
109031397653295178343967849129618747730638849
27.128209540870
$c$:
109031397653295180407229810953140258789062500
27.110589165866
$a$:
6383506767777189291700242535849759902342269604947081
27.110589165866
$b$:
4029415384727100446779695778840655465861318710937500000
27.110589165866
$c$:
4035798891494877636071396021376505225763660980542447081
27.052986018345
$a$:
653183680307783067
27.052986018345
$b$:
1399554597612075731651306152343750
27.052986018345
$c$:
1399554597612076384834986460126817
26.962005369793
$a$:
246674545293823961128566332
26.962005369793
$b$:
76176897712118105169434411798376043
26.962005369793
$c$:
76176897958792650463258372926942375
26.926561559958
$a$:
720077960137146368
26.926561559958
$b$:
121494451649631319056757
26.926561559958
$c$:
121495171727591456203125
26.909403452311
$a$:
39590256758143909424533741609695181914455679917839575312452349251518764036737929216
26.909403452311
$b$:
291550496198577256536957307136564810380957037620342264865157097947944616551250527735164675371741649152587890625
26.909403452311
$c$:
291550496198577256536957307176155067139100947044876006474852279862400296469090103047617024623260413189325819841
26.874321560129
$a$:
1025227
26.874321560129
$b$:
191008735430507276015771648
26.874321560129
$c$:
191008735430507276016796875
26.862823384503
$a$:
24833
26.862823384503
$b$:
5020969537415167
26.862823384503
$c$:
5020969537440000
26.860251070472
$a$:
14635787791951924463179043
26.860251070472
$b$:
46705984699784297218048531659032073
26.860251070472
$c$:
46705984714420085010000456122211116
26.800594498786
$a$:
429280172117791132385483537443684438408457
26.800594498786
$b$:
2902879259661729548771784436643414887366656
26.800594498786
$c$:
3332159431779520681157267974087099325775113
26.775976643862
$a$:
226528683441437300000
26.775976643862
$b$:
61205917571030987668745001
26.775976643862
$c$:
61206144099714429106045001
26.759298895452
$a$:
2384990205200861462116854028584329
26.759298895452
$b$:
5037698588038241041943612289487253637431296
26.759298895452
$c$:
5037698590423231247144473751604107666015625
26.743097878946
$a$:
1166734795934435022801
26.743097878946
$b$:
37173286594669935706660000000
26.743097878946
$c$:
37173287761404731641095022801
26.728372140691
$a$:
388365712944346688648070164763973109755796355853949
26.728372140691
$b$:
24411408707750782855607390395369452215770875147736026890244203644146051
26.728372140691
$c$:
24411408707750782855995756108313798904418945312500000000000000000000000
26.704089935634
$a$:
162729
26.704089935634
$b$:
1042151581574097478352896
26.704089935634
$c$:
1042151581574097478515625
26.628873238595
$a$:
6646061257
26.628873238595
$b$:
575964977644707347858848400000000000000
26.628873238595
$c$:
575964977644707347858848400006646061257
26.622447269428
$a$:
31117999167337103924704
26.622447269428
$b$:
75929189120022207165685625
26.622447269428
$c$:
75960307119189544269610329
26.541902118184
$a$:
10888990400037112340748947
26.541902118184
$b$:
313598822099962887659251053
26.541902118184
$c$:
324487812500000000000000000
26.527526272782
$a$:
1220490749793073
26.527526272782
$b$:
1293730089605146013383786496
26.527526272782
$c$:
1293730089606366504133579569
26.526815642959
$a$:
11529295992324189492061
26.526815642959
$b$:
850844577839153422257109336064
26.526815642959
$c$:
850844589368449414581298828125
26.525254340693
$a$:
218312028734669511361729
26.525254340693
$b$:
406680828357186701864265471296308414726679922999296
26.525254340693
$c$:
406680828357186701864265471514620443461349434361025
26.516336519896
$a$:
15882708203729084657
26.516336519896
$b$:
8095815186379034155707810947817231
26.516336519896
$c$:
8095815186379050038416014676901888
26.509833550350
$a$:
664559691245401291839706490968570625
26.509833550350
$b$:
4051734037392610655275387090022711296
26.509833550350
$c$:
4716293728638011947115093580991281921
26.471884373519
$a$:
297821847037375790182048338974904789595625
26.471884373519
$b$:
1408449959449829865903874434854235561618770503906729026505861496832
26.471884373519
$c$:
1408449959449829865903874732676082598994560685955068001410651092457
26.464164935221
$a$:
454706363197252713600001231975318360612675868588903908546411045644584041
26.464164935221
$b$:
1028673825701074701587123624563195756962445311130745382010936737060546875000000000000000000000
26.464164935221
$c$:
1028673825701074701587578330926393009676045312362720700371549412929135778908546411045644584041
26.447619024058
$a$:
5482474433575759854689047243627307307663828298730881
26.447619024058
$b$:
2427666149827775581474264951015301706957873433664159744
26.447619024058
$c$:
2433148624261351341328953998258929014265537261962890625
26.397863532805
$a$:
318306095337360132090430620573027032629473861845141946368
26.397863532805
$b$:
353869912946906944901308235078781710425035644286207663007
26.397863532805
$c$:
672176008284267076991738855651808743054509506131349609375
26.299983366554
$a$:
51064811
26.299983366554
$b$:
7826989785439209430125504708476928
26.299983366554
$c$:
7826989785439209430125504759541739
26.299271926642
$a$:
16410601
26.299271926642
$b$:
324209127189310500467198327318538505890963105024
26.299271926642
$c$:
324209127189310500467198327318538505890979515625
26.281583252583
$a$:
1628995676032061914468351888326691
26.281583252583
$b$:
151313210503692653503645417054628785562845184
26.281583252583
$c$:
151313210505321649179677478969097137451171875
26.268406328623
$a$:
47083785239352742984619140625
26.268406328623
$b$:
311958186094982909351657447424
26.268406328623
$c$:
359041971334335652336276588049
26.233016114317
$a$:
1064688249377365
26.233016114317
$b$:
9509162922100286472565114284547861931
26.233016114317
$c$:
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3060767416910056002654667803542453269473
24.513684992322
$a$:
31261091101632060539508193
24.513684992322
$b$:
12350667342087531953099217904903692170432
24.513684992322
$c$:
12350667342087563214190319536964231678625
24.506186932190
$a$:
194478079441586388803257968359375
24.506186932190
$b$:
4836408215518349253440286038941680046129
24.506186932190
$c$:
4836408409996428695026674842199648405504
24.489573112792
$a$:
6044112236696882681065850397930035
24.489573112792
$b$:
184722922760676649502680217573144339254388748010044493
24.489573112792
$c$:
184722922760676649508724329809841221935454598407974528
24.487462058206
$a$:
674319184954167467886814321451800429
24.487462058206
$b$:
32072945385497021757428645032113185678548199571
24.487462058206
$c$:
32072945386171340942382812500000000000000000000
24.486567807777
$a$:
82800367216340571
24.486567807777
$b$:
78295316104448477385125
24.486567807777
$c$:
78295398904815693725696
24.457380865250
$a$:
116051814817175188995924281418394057836364267701
24.457380865250
$b$:
1168903597518928984229368851478883329604266909211232305152
24.457380865250
$c$:
1168903597634980799046544040474807611022660967047596572853
24.447207018833
$a$:
210843112500000000000000
24.447207018833
$b$:
151621931292374320157056076023
24.447207018833
$c$:
151622142135486820157056076023
24.439748295897
$a$:
374767263669735324912532848758465179
24.439748295897
$b$:
27196791441961265221185690344499359288954918384552001953125
24.439748295897
$c$:
27196791441961265221186065111763029024279830917400760418304
24.431573196836
$a$:
12887364131254655099910734519749499397
24.431573196836
$b$:
22305047981483218559409661579858523593970061897851450500603
24.431573196836
$c$:
22305047981483218559422548943989778249069972632371200000000
24.408287410413
$a$:
100392141091355154507792843
24.408287410413
$b$:
3259846319543580408464554373004099125
24.408287410413
$c$:
3259846319643972549555909527511891968
24.354404971834
$a$:
6346605553
24.354404971834
$b$:
492975898921684287283512668474514447
24.354404971834
$c$:
492975898921684287283512674821120000
24.334374567560*
$a$:
40088871825669269806676699404398438617069570145882198913022304508970275069
24.334374567560*
$b$:
10297818852381490627980605609794739794565468993810712757163364188160000000000000000000000000000000000000000
24.334374567560*
$c$:
10297818852381490627980605609794779883437294663080519433862768586598617069570145882198913022304508970275069
24.327087603887
$a$:
3726611675441373705930335953354752
24.327087603887
$b$:
149396799168455271303955954571533203125
24.327087603887
$c$:
149400525780130712677661884907486557877
24.302509922005
$a$:
12107308913642286980768214025994818038644498026458433249
24.302509922005
$b$:
123676029973211471508626542891385793722839330765833024605289917961355501973541566751
24.302509922005
$c$:
123676029973211471508626542903493102636481617746601238631284736000000000000000000000
24.284687194403
$a$:
38035127243
24.284687194403
$b$:
150815359125848074407376393311363522682778682421
24.284687194403
$c$:
150815359125848074407376393311363522720813809664
24.278027424448
$a$:
141738889709
24.278027424448
$b$:
6489879932415727624554342702727592889600
24.278027424448
$c$:
6489879932415727624554342702869331779309
24.277629384941
$a$:
2177034288120125
24.277629384941
$b$:
347345003492078554423324947106496512
24.277629384941
$c$:
347345003492078554425501981394616637
24.271411867580
$a$:
107765161
24.271411867580
$b$:
16814226186045162242785754358008663
24.271411867580
$c$:
16814226186045162242785754465773824
24.260852045942
$a$:
1150214810701824
24.260852045942
$b$:
27270489383229770695157551
24.260852045942
$c$:
27270489384379985505859375
24.256343568827
$a$:
44
24.256343568827
$b$:
14087010572580078081
24.256343568827
$c$:
14087010572580078125
24.249684553133
$a$:
22179156666895547093
24.249684553133
$b$:
34913183682284144146198407675904
24.249684553133
$c$:
34913183682306323302865303222997
24.239399559325
$a$:
557832031250
24.239399559325
$b$:
126612145087527382220763
24.239399559325
$c$:
126612145088085214252013
24.214906050883
$a$:
946405371868870625
24.214906050883
$b$:
32374802723214377186364016433232
24.214906050883
$c$:
32374802723215323591735885303857
24.214278760493
$a$:
26323433299859116561308330229760000
24.214278760493
$b$:
2468757154125328622430804067288873801
24.214278760493
$c$:
2495080587425187738992112397518633801
24.205506823245
$a$:
22085158925511
24.205506823245
$b$:
20716509203959465726291916489
24.205506823245
$c$:
20716509203959487811450842000
24.201029361597
$a$:
17449402268886407318558803753801
24.201029361597
$b$:
2481883971970720407896821114796246199
24.201029361597
$c$:
2481901421372989294304139673600000000
24.200804025757
$a$:
6076542700690680970287890694505190153107603050839637477768938585102002911740657261
24.200804025757
$b$:
9269874296098644349859031238354610813235430776263605108213374933834870534152350370951875
24.200804025757
$c$:
9269880372641345040540001526245305318425583883866655947850852702773455636155262111609136
24.197559656696
$a$:
12893223714967486449319936
24.197559656696
$b$:
1282060125929359208099325277659184767755859375
24.197559656696
$c$:
1282060125929359208112218501374152254205179311
24.174749236072
$a$:
70481196916823187
24.174749236072
$b$:
2624990993600261964900937229950772705078125
24.174749236072
$c$:
2624990993600261964900937300431969621901312
24.156771188339
$a$:
119529069824030642258141
24.156771188339
$b$:
24065179596584182939170785357741859
24.156771188339
$c$:
24065179596703712008994816000000000
24.126954066950
$a$:
420707233300201
24.126954066950
$b$:
2095497085117617283137815
24.126954066950
$c$:
2095497085538324516438016
24.115686466089
$a$:
3211838877954855105157369
24.115686466089
$b$:
12209871893875847187098385907192246272
24.115686466089
$c$:
12209871893879059025976340762297403641
24.110754254342
$a$:
27027732776139404140491414984430327896517325
24.110754254342
$b$:
824510321926420065813537737857678172568620732043535537459
24.110754254342
$c$:
824510321926447093546313877261818663983605162371432054784
24.071925833406
$a$:
613737032673817
24.071925833406
$b$:
645918268034832102287372054052905524096
24.071925833406
$c$:
645918268034832102287372667789938197913
24.016847205244
$a$:
25042000461254908942243489
24.016847205244
$b$:
746238520246620316957147136
24.016847205244
$c$:
771280520707875225899390625
Definition
The list contains all known $abc$-triples merit at least $24$, as givin in [2]. An $abc$-triple is a solution of the equation $a+b=c$ with coprime integers $0 < a \leq b < c$. The quality of an $abc$-triple is defined as $q = \log(c)/\log(r)$, where $r = \text{rad}(abc)$ is the radical of $abc$, that is, the product over all prime divisors of $abc$. The merit of an $abc$ triple is defined as $m = (q-1)^2 \log(r) \log\log(r)$.
Parameters
$m$
—   real number ($m > 24$)
Comments
(1)
The abc-conjecture [4] states that $\limsup q = 1$, that is, for every $\varepsilon > 0$ there are only finitely many $abc$-triples of quality $q > 1+\varepsilon$.
(2)
The refined $abc$-conjecture of Robert, Stewart and Tenenbaum [1] implies that $\limsup m = 48$.
References
[1]
O. Robert, C. Stewart, G. Tenenbaum, A refinement of the abc conjecture, Bull. Lond. Math. Soc. 46:6 (2014), 1156-1166. https://hal.archives-ouvertes.fr/hal-01281526/document
Links
Data properties
Numbers are of type: integer
Table is complete: unknown
Sources of data: [3] and the references therein
Reliability: The given merit $m$ might not be reliable, in which case it is signified with an asterisk '*': By definition, $m$ depends on the prime factorization of $abc$, which might be out of reach. If this is the case, the large remaining factor of $abc$ is heuristically assumed to be square-free, such that a heuristic value for $r$ (and thus for $q$ and $m$) can be deduced.