Markov quadratic irrationals

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Numbers

$m_1$, $m_2$, $m$

$\xi_{m_1,m_2,m}$

1, 1, 1:

1.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137

equals: Golden_ratio#phi
comment: This entry is the golden ratio $\varphi$.

1, 1, 2:

2.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573

equals: Algebraic_numbers_of_degree_2#1,-2,-1,2
comment: This entry is $1+\sqrt2$.

1, 2, 5:

2.386606874731850552261200821393139665144898551372086156056309481025183731478116765861583646027396110

1, 5, 13:

2.382641699550034350001685085928543935263766480365666839454878554943389785602559883628950924339314494

2, 5, 29:

2.413396697521692069573909262471775360405089694897582145298036786854911941222653458610819114739030922

1, 13, 34:

2.382064562821407302949477679124347643989344395911646002230873766007627538841830645761279073851156473

1, 34, 89:

2.381980389091116310713462596070053713492205115409806087378616477876112863924675647609587444299623833

2, 29, 169:

2.414189512459993941377869066991467583023790111759138649606156863363895915324731620068614609687379819

5, 13, 194:

2.386589081349839926790985609831335821495482482881382547042930464188345661005543508647960216741779654

1, 89, 233:

2.381968108935225256641113074812140885279749752906811711512706194503752942278725801916863419878243222

5, 29, 433:

2.413393141285457408056864136906117089768151617026602863250492866411772071975695517081379380296793630

1, 233, 610:

2.381966317297949829176217510251809600474260203605811200041930284889748161552460791022722927440363151

2, 169, 985:

2.414212854406584547914379028107552673495891061731120055154651236047783214964367715508961776221869111

13, 34, 1325:

2.382641319568043044757328539210588338000535183148350044348793625948286745541760397700190377507674488

1, 610, 1597:

2.381966055901877755554644868544962727182460155423625474928947020053898495720542981138871192011293395

5, 194, 2897:

2.386606794939054381493587010756283291195287479888721038114265742054020902089294206097806895399169555

1, 1597, 4181:

2.381966017764710846852448275927912511182418605368637251637945515144037416357291039683876446225510806

2, 985, 5741:

2.414213541532504062929995376628208249555308779567959405358226988545112788849411759276920405882287993

5, 433, 6466:

2.413393125339977343259624415716005216396806638284314418649116037190591662788309090539280127374755762

13, 194, 7561:

2.386589069688416958266597331667141559051852841886939853155425508124682113680439426643438463647043178

34, 89, 9077:

2.382064554729221914650693884472006476360303766303302687672418503731008675205669688904371064519864906

1, 4181, 10946:

2.381966012200573310568009047629930478700962812162110464004163934069709929917721497879360919324282871

29, 169, 14701:

2.414189509375267362023151426499309423609823909202648356965155427255444596002758429751595977007658911

1, 10946, 28657:

2.381966011388776587243126078971326343033721404849367026661435634082672514421125887672201358410457241

2, 5741, 33461:

2.414213561759605327685565765940024093118032606363274017619502245591742753433519044575348960260826378

29, 433, 37666:

2.413396697051725318458930992333081901178053227586681098173299281092493978197819808974517981950736798

5, 2897, 43261:

2.386606874374028122359476863482127992209191836730223931241566345064635704504824875891143024059336170

13, 1325, 51641:

2.382641699299881499365971359480252040801523165680616450640998584579372400762149165053671114436924448

89, 233, 62210:

2.381980388918852349140706420521502274983285100697320912324872201251392142739234193487617931802901273

1, 28657, 75025:

2.381966011270337041563301925799464035542052973070616842397090259084382573060600079836100466511916840

5, 6466, 96557:

2.413393125268471552401476734922475126682175760013065289202370042592646351046576987516721753505807924

34, 1325, 135137:

2.382641319531537383343189357029172721463408347895032733242156881116940148430161695879773121979673472

2, 33461, 195025:

2.414213562355035597572442159223406978568250707624962233518300251890260789656805549220418183685196542

1, 75025, 196418:

2.381966011253056944731574306075639257672644788495605096156431326649844283936709658047127949315270723

13, 7561, 294685:

2.386589069680739938095270193943191196078380739876727676772215245943736477008648202645454174112426861

233, 610, 426389:

2.381968108931558368640045674980518191805614293302259833642141037266583418494467309066379960250425156

169, 985, 499393:

2.414212854403911394465430620605678590920032784943541899110265892578804330933441211345574702406923311

1, 196418, 514229:

2.381966011250535812580836572669054523922813904065909813850020063461498069612779445812946857319292879

5, 43261, 646018:

2.386606874730245935105771346242018535260559162694364003899641393026746189140748802531338085160192658

34, 9077, 925765:

2.382064562820629357797198797356343821806674841735703710614640965056219172322057006729103965382553235

2, 195025, 1136689:

2.414213562372563428193486793285105457216328580262989798940814000137603540326598784850237557084840434

29, 14701, 1278818:

2.414189509374859708443692188354997429060625532727362634525484866796024927703881148377982401582520436

1, 514229, 1346269:

2.381966011250167984357219795022430592843332725923545102518047987325392633297023304132901585327264161

5, 96557, 1441889:

2.413393125268150892183373770347914428115492724251074183432184572262805269512292702341352667257387927

194, 2897, 1686049:

2.386606794938819867715798237013415486753119926822848677060159836375233077202560517982372317772068046

13, 51641, 2012674:

2.382641699549869667342658490813356526716035722900006335683946816125597604050737715758845224985961344

89, 9077, 2423525:

2.382064554729108409979904506697144256616829963606137684584176142190567692171653563936225617257078676

610, 1597, 2922509:

2.381966317297871774818198351891586476105200080257907760491157608583785003005268031771550589058535252

29, 37666, 3276509:

2.413396697521629962165509701840052370755626340765267476962279491431014816292408674981828098450317366

1, 1346269, 3524578:

2.381966011250114318942636247770680649815684597232325772658212773764394784085339077973460850271959519

194, 7561, 4400489:

2.386589081349805499080381364776097360300808050754174378492363311784002551102680511985267650603346151

2, 1136689, 6625109:

2.414213562373079399352130346921845916506576597252040816727509815240338881628058803908824397854211753

169, 14701, 7453378:

2.414189512459981940745373891588284384762409728175635813028254311568807628042584332862665732249865452

433, 6466, 8399329:

2.413393141285447958317882912353091483386288827799144522184690172524905171462330872333974462100516194

1, 3524578, 9227465:

2.381966011250106489264168112976957395630171994338629413204359664784453188194063360248447566110837987

5, 646018, 9647009:

2.386606874731843356523873650077081623233637333786825281582472503956984947557876233091527404460340668

13, 294685, 11485154:

2.386589069680734884096926800043131243811613870658681132042233423707417848251726651724887152784019652

34, 135137, 13782649:

2.382641319531533873858528563400813288229270679073279637979695231801978614902046310239554829926247455

89, 62210, 16609837:

2.381980389091113894228526891099441510427550718688794425496600360619321122611208160182919569070885138

985, 5741, 16964653:

2.414213541532501746502129906378262491832667204253353279683196184573770538782585106142895386702427854

1597, 4181, 20031170:

2.381966055901876094070786393850656387946004172827133452769604512193877501938617501540046672588489825

5, 1441889, 21531778:

2.413393125268149454216259367317480088170096545634840932650816318080871828537668893616689049036688861

1, 9227465, 24157817:

2.381966011250105346929474714934025277553089053298834575519447089183265809467410112323733985541611124

2, 6625109, 38613965:

2.414213562373094588124905879954949419667352368016021414749416121864036063338240440347957425280886363

233, 62210, 43484701:

2.381980388918851996578492624681976321969111672109812956409349140471080444528697621165154841086822830

433, 37666, 48928105:

2.413396697051725039980252398626145323027064851606383857079350644736684486632655515460545707552662627

1, 24157817, 63245986:

2.381966011250105180265089063390215044413848866627641576730889856474180837533283226640310605424834248

13, 2012674, 78442645:

2.382641699550034241586457699889152275090625536482458792536959677603201376389096641661796485170931484

34, 925765, 94418953:

2.382064562821407228161440814217841054000115093687511863821041781160436707962199852493924645385013597

29, 1278818, 111242465:

2.414189509374859654571142119752137487684983277228176288190241403140522530874371768653893630318442668

4181, 10946, 137295677:

2.381966017764710811485709238410829728402243138480158652872710280559106956420893437129713012813034364

5, 9647009, 144059117:

2.386606874731850519992671529884953240811679549285341560942209011659130807081667851677370731745746674

1, 63245986, 165580141:

2.381966011250105155949082900625687755419399486317057300101395578597155024119700673576922230775852375

1325, 51641, 205272962:

2.382641699299881483544559254558019436086534903127826993307780263236927934796635220367122948479973410

2, 38613965, 225058681:

2.414213562373095035240630396881683075158059776138304547171951127635283659624688229268039922933600022

169, 499393, 253191266:

2.414212854403911384065958372888169231818454951204683145661989844760105585082755062201571647533936507

29, 3276509, 285018617:

2.413396697521692061366243148526077646566507154441679030293160142975053330358712629856796746229472603

233, 426389, 298045301:

2.381968108935225249136210279094178084325786291227384945249636063145176452139734445579158299426160563

5, 21531778, 321534781:

2.413393125268149447767846669410162439904829656760023200124203671595125366333402167903820830089689555

2897, 43261, 375981346:

2.386606874374028117643451367178737118258483899488849938254518193125380261020117646462808926187872214

1, 165580141, 433494437:

2.381966011250105152401425412817790196830668833909484514116826107410965659955380838103402754803609884

13, 11485154, 447626321:

2.386589069680734880769737508555471595091175218275496801923321370656815715018630208656784378025035795

1325, 135137, 537169541:

2.382641319568043042446934739272355585286754204770177142316786247597643509019277327422966326299617363

5741, 33461, 576298801:

2.414213561759605325678261031462595666364302719087282374612556209779010559528214542251119639490522278

89, 2423525, 647072098:

2.382064554729108408387683335113984942004825109551852407420571821968143473008105797198458656904699729

610, 426389, 780291637:

2.381968108931558367545094292116837937386866573486506411602007600656614812998161347636109492510823703

10946, 28657, 941038565:

2.381966012200573309815184281600921614942251275500773911372838327079159088445513728688702423980318149

194, 1686049, 981277621:

2.386606794938819867023449423868505873445422207408499940754094285210135827562953329675719904316100797

1, 433494437, 1134903170:

2.381966011250105151883829160927034218760071348460370069710443815079280429131311706325078346181949919

2, 225058681, 1311738121:

2.414213562373095048402488439311976125178597823760774732000853895175979046510148597060229995249132741

34, 13782649, 1405695061:

2.382641319531533873521142993971187947963508755511944699530749371588567120343617968116982907729085433

985, 499393, 1475706146:

2.414212854406584547608246823811980671542559303980079462619984704154115373171561213638033853869886087

6466, 96557, 1873012681:

2.413393125339977343069592167386199293144161454784490508520480242620404119587457113279812598539228062

5, 144059117, 2151239746:

2.386606874731850552116495986392409476267230814580368492695128129030420019558953895108673237110819718

194, 4400489, 2561077037:

2.386589081349839926689345590569093268825718130666737377916414029009894631508741664213441540870209808

1, 1134903170, 2971215073:

2.381966011250105151808312885499639923256626731017408941902853453425698682261415748044345614338608910

13, 78442645, 3057250481:

2.382641699550034349930312291739763349835463754009409969620631453222280582792865314824568374988443560

169, 7453378, 3778847945:

2.414189512459993941331182536919702288659490902087011784216385994734440781825144105180308527996375804

89, 16609837, 4434764269:

2.381980389091116310679564621177289454888756661958677805711749731081368545666413772939226233998016517

5, 321534781, 4801489937:

2.413393125268149447738929439032611539015043791943021254689341958154905585997023726704650367946025536

610, 2922509, 5348189873:

2.381966317297949829152910034772064820136683583655383309470176783796759935968520778577915558324443587

28657, 75025, 6449974274:

2.381966011388776587227101273134541063321556670076768357666034764257795065076105265175465439830412425

7561, 294685, 6684339842:

2.386589069688416958251676545795871280426966914506888696215956045033970833717523244941320659596274721

2, 1311738121, 7645370045:

2.414213562373095048789937365015166700916768890045153612143450878837970192914802888385948890801788320

1, 2971215073, 7778742049:

2.381966011250105151797295209398635832641379223056052241452215940441095283493948033659005306777899123

34, 94418953, 9629807441:

2.382064562821407302942287904316032118777600720137106644710355689959153051100695877873130186796245545

29, 111242465, 9676815637:

2.414189509374859654564022712984667085645607631422619625028684670358973701529530725740313238888664549

433, 8399329, 10910721905:

2.413393141285457408051263956402734139598872909845445737575223393940608686508163041727788122679431577

1597, 2922509, 14001740009:

2.381966317297871774814797836675194147038840453889204223852463834617146385745897107244330118034130194

2897, 1686049, 14653451665:

2.386606794939054381490482244751942580897917839035063394431431264294643503653368196040547003854353680

13, 447626321, 17445941365:

2.386589069680734880767547126242451216798384888022100581393526281946896686305694139239432567448635193

33461, 195025, 19577194573:

2.414213562355035597570702725737026084389599457169882360853380071488765927897615240194540218391061394

1, 7778742049, 20365011074:

2.381966011250105151795687752119001493834648604918316538256688863397656541338887359324352657451488321

29, 285018617, 24793343170:

2.413396697521692069572824596652125923203894152934003273529183421863214138193166512707296704971182414

9077, 925765, 25209506681:

2.382064562820629357796149786345757200218085460999057758513715817368600046346304292400486329037812618

233, 43484701, 30395743789:

2.381980388918851996577771046836366334544217481659569016032561930745071567251505683713199822518018923

5, 2151239746, 32124537073:

2.386606874731850552260551907738494146410644980841491943100080783411788092861709553780170062710336011

75025, 196418, 44208781349:

2.381966011270337041562960817996493606730073469267530516259202414959026715364811249850452076976149304

2, 7645370045, 44560482149:

2.414213562373095048801342796493353191791708228463929886007734615485877092607766102600306831829184597

985, 16964653, 50130543874:

2.414213541532501746501864626745162444044465000451488463164190381924720881768047634609955586336185860

1, 20365011074, 53316291173:

2.381966011250105151795453227262565212802708504218869957745146390932898174443438809752084367155554313

14701, 1278818, 56399710225:

2.414189509375267362022941844009263957790153882774462611060545765067623706219093154158573484168392749

433, 48928105, 63557570729:

2.413396697051725039980087364346421324714278141793166308716826242547968743160602716085473032762264688

9077, 2423525, 65995009186:

2.382064554729221914650540815715457225529105457444520134453555306802125042679362858234661869709865492

5, 4801489937, 71700814274:

2.413393125268149447738799762746076757503311656093233762461908675099710452102539369458028974882003308

43261, 646018, 83842154089:

2.386606874730245935105676507914747798498198099540981135140970881721304703352466444502009260927652302

1597, 20031170, 95969331289:

2.381966055901877755554572484337687078781176951191654194117810884126017156011782806004436117708670241

7561, 4400489, 99816291793:

2.386589081349805499080314452488523471457797188498301964463586595808701956654999034341634012568038657

13, 3057250481, 119154326114:

2.382641699550034350001638099210228293979736517942560779505108521419588503967706744059304610195972320

169, 253191266, 128367472469:

2.414212854403911384065917915414447032106116015627833130520067566613713457965705920523590929239231648

1, 53316291173, 139583862445:

2.381966011250105151795419010547145584385856857568971671314757682522746016981264424683777225620834949

34, 1405695061, 143367113573:

2.382641319531533873521110559288189702909441915730046205594195366454513137044075623790022553719135436

6466, 8399329, 162930183509:

2.413393141285447958317857798945366813659440403873405916112972775321005670002069557404962874417815820

89, 647072098, 172765826641:

2.382064554729108408387660999748884763216284665714377301012778687840667335567724948770161665805864531

233, 298045301, 208333239010:

2.381968108935225256641097714766795419161878915089885676922725655130830122441697755823174876183628918

4181, 20031170, 251250963713:

2.381966055901876094070775833136971644184387994839444486588997253863971769558749662865515374092857846

2, 44560482149, 259717522849:

2.414213562373095048801678541048503305771836570887117866378000957422236286238194420073089513288175447

5741, 16964653, 292182217634:

2.414213541532504062929987567524681470537276126578194898912133148926388081204022588892624590836148142

196418, 514229, 303011495165:

2.381966011253056944731567045174357655415353047116114927460428804297969361989981033227847713379624110

51641, 2012674, 311809494089:

2.382641699549869667342651633877738988679455629992820922679744073329959033968240388392357598166894970

14701, 7453378, 328716329765:

2.414189512459981940745367721857948356896397044598430849283250381835327763460264251326228077407367054

1, 139583862445, 365435296162:

2.381966011250105151795414018395644466499835398514911081286502029101570242490272188615319952194020949

37666, 3276509, 370238963953:

2.413396697521629962165504838389586137549591859460342379939438570068324482452834551152488781715889422

96557, 1441889, 417673428514:

2.413393125268471552401472913412069870463998016832156901409609744687946948908131021999587988514628427

5, 32124537073, 479716816349:

2.386606874731850552261197911407919176287572504847093486731778821858851447043188581911449618802971866

194, 981277621, 571101889373:

2.386606794938819867023447379865420302041708200671253479040370550185368994181311375783097450345256812

195025, 1136689, 665048316673:

2.414213562372563428193485285975926444627521794420166939817479559788132472504928990672832313515037257

13, 17445941365, 679944086914:

2.386589069680734880767545684252153439867356512610432308639295259584474420200921371031766202419498223

1325, 205272962, 815959972309:

2.382641699299881483544558253242305222596234200776721743017870632851586656159995321675584023598556517

29, 9676815637, 841771717954:

2.414189509374859654564021772135335539954032818504354338258343320666460834348048540333825839372335427

1, 365435296162, 956722026041:

2.381966011250105151795413290050556269714536831079592383791592794742759986178766553244680868826759873

34, 9629807441, 982145940029:

2.382064562821407302949476987932761913987732121784085575056041708507250450287447667075682495351973753

Definition

For a Markov triple $(m_1,m_2,m)$ [5] with $m_1\leq m_2\leq m$, let $k$ be the unique integer with $0\leq k\leq m/2$ and $k m_2\equiv\pm m_1\pmod m$ [1]. This table stores $\xi_{m_1,m_2,m}=((2k+m)+\sqrt{9m^2-4})/(2m)$.

Parameters

$m_1$

— smallest member of the Markov triple ($m_1\geq 1$)

$m_2$

— middle member of the Markov triple ($m_1\leq m_2$)

$m$

— largest member of the Markov triple ($m_2\leq m$ and $m_1^2+m_2^2+m^2=3m_1m_2m$)

Formulas

(1)

$m x^2-(2k+m)x+(l+k-2m)$, where $l=(k^2+1)/m$. The stored value $\xi_{m_1,m_2,m}$ is the larger root of this polynomial.

(2)

$(2k+m)^2-4m(l+k-2m)=9m^2-4$, the discriminant of the quadratic irrational.

(3)

$L_m=\sqrt{9m^2-4}/m$ is the corresponding element below $3$ of the Markov and Lagrange spectra.

Comments

(4)

Cassels's Markov form for the same triple is $F_C(x,y)=m x^2+(3m-2k)xy+(l-3k)y^2$, where $l=(k^2+1)/m$. The root of $F_C(x,1)=0$ is $\xi_{m_1,m_2,m}-2$, so the two normalisations differ by the integer $2$.

(5)

Along the Fibonacci branch $(1,F_{2n-1},F_{2n+1})$, where $F_n$ denotes the $n$th Fibonacci number, the values approach $4-\varphi$, where $\varphi$ is the golden ratio. Along the Pell branch $(2,P_{2n-1},P_{2n+1})$ for $n\geq 2$, where $P_n$ denotes the $n$th Pell number, they approach $1+\sqrt2$. These are not the only visible branch limits in the stored range: along any branch for which $k/m$ converges, the defining formula gives the limit $2+\lim k/m$.

(6)

Large Markov triples give values that agree with their branch limit in many leading digits. This is the same crowding near the bottom of the Markov and Lagrange spectra [4].

(7)

In the $\mathrm{GL}_2(\mathbb Z)$-equivalence class of quadratic irrationals attached to the Markov triple [1], the stored value is the representative in $(1,3)$. Its regular continued fraction is purely periodic, with every partial quotient in $\{1,2\}$ [10].

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.arith.misc import inverse_mod
from sage.rings.integer_ring import ZZ
from sage.rings.real_arb import RealBallField


def markov_xi(m1, m2, m, digits=100):
    m1, m2, m = ZZ(m1), ZZ(m2), ZZ(m)
    if not (m1 <= m2 <= m and m1*m1 + m2*m2 + m*m == 3*m1*m2*m):
        raise ValueError("expected a normalised Markov triple")
    if m == 1:
        k = ZZ(0)
    else:
        inv = inverse_mod(m2, m)
        candidates = {(m1 * inv) % m, (-m1 * inv) % m}
        choices = [k for k in candidates if 2*k <= m]
        if len(choices) != 1:
            raise ValueError("expected one k with 0 <= k <= m/2")
        k = choices[0]
    R = RealBallField(numberdb.bits(digits, losing=32))
    return (R(2*k + m) + R(9*m*m - 4).sqrt()) / R(2*m)


print(markov_xi(1, 1, 1))
print(markov_xi(1, 2, 5))

References

[1]

J. W. S. Cassels, An Introduction to Diophantine Approximation, Cambridge University Press, 1957, Chapter II.

[2]

M. Aigner, Markov's Theorem and 100 Years of the Uniqueness Conjecture, Springer, 2013. (doi)

[3]

O. Perron, Ueber die Approximation irrationaler Zahlen durch rationale, II, Sitzungsberichte der Heidelberger Akademie der Wissenschaften, 1921, 8. Abhandlung, 1-12.

Links

[4]

Wikipedia: Markov spectrum

[5]

Wikipedia: Markov number

[6]

MathWorld: Markov Spectrum

[7]

MathWorld: Regular Continued Fraction

[8]

OEIS A291694: array of Markov triples sorted by largest member

[9]

OEIS A305310: Cassels's $k$ for Markoff forms

[10]

OEIS A305317: periods of continued fractions of Markoff-form quadratic irrationals

[11]

OEIS A002559: Markov numbers

[12]

OEIS A305313: smaller member of the Markov triple

[13]

OEIS A305314: middle member of the Markov triple

Similar tables

Golden ratio — the first entry is $\varphi$, and the Fibonacci branch accumulates at $4-\varphi$

Algebraic numbers of degree 2 — holds quadratic algebraic numbers by their defining polynomial rather than by Markov triples

Khinchin's means $K_p$ — almost-sure regular-continued-fraction limits, in contrast with these extremal purely periodic continued fractions

Lévy's constant — the almost-sure denominator growth rate for generic regular continued fractions, in contrast with the extremal purely periodic expansions attached to Markov triples

Lochs's constant — the almost-sure conversion rate between decimal digits and regular continued-fraction partial quotients, in contrast with the extremal purely periodic expansions attached to Markov triples

Eigenvalues of the Gauss-Kuzmin-Wirsing operator — spectral data for the Gauss map governing generic regular continued fractions, in contrast with the extremal purely periodic expansions attached to Markov triples

Hermite's constants — the corresponding extremal constants for definite quadratic forms

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds every normalised Markov triple with largest member $m\leq 10^{12}$)

How they were obtained:

Each value is an explicit quadratic irrational. The generator computes $\sqrt{9m^2-4}$ in real ball arithmetic from the exact integer discriminant and returns the resulting real ball, so the stored decimal interval follows from the width of the ball.

Before the entries were written, the generator was checked on the full stored range against the Markov equation, the uniqueness of $k$ in $0\leq k\leq m/2$, the quadratic equation in Formula (1), the discriminant identity in Formula (2), and the purely periodic regular continued fraction property. The first 40 largest Markov numbers were compared with OEIS A002559 [11]; their two smaller members were compared with OEIS A305313 and A305314 [12] [13]; and the resulting first 23 triples were compared with the rows exposed by OEIS A291694 [8]. The first 40 $k$ values were compared with OEIS A305310 [9], and the first 40 period lengths were compared with OEIS A305317 [10].