Lagrange numbers $L_m$ of the Markov spectrum

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Numbers

$m$

$L_m$

2.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275

2.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145

2.973213749463701104522401642786279330289797102744172312112618962050367462956233531723167292054792220

13:

2.996052629869299469234139402626318639758302191500564448140526340656010340435888998027132617909398219

29:

2.999207188146832415009887490460792100120524217381371187147797711640858365203927606876810643271165293

34:

2.999423243289873429428367122954577640919865262411527298579394590838784489448367173875499324172901181

89:

2.999915834362007902325801596634489449456320343179162736554985764740989772793171519938276012194753285

169:

2.999976658056082557311951151734414455988408625885141204537757513710040351359522411734862355469434195

194:

2.999982286411020059767538229971949993506428883288538289962149588170505755000777738945404969566033534

233:

2.999987720016373260063342029452607950817010235427357328604811530638407172111099672503254736752194599

433:

2.999996444233732368076776772657499768450852887632882400865881806726546453420213205063451601240238530

610:

2.999998208366391461631123545093783135374749915408343711559270405845070093596724860733970445044660729

985:

2.999999312874082801412514401392770321611071463563762952948896380725008054294217664520461623509728070

1325:

2.999999620268161561212770285968346487321825083277832164169285365104120660894841550117361887090820675

1597:

2.999999738604005980739847032017915435579698019050131350609302931779681775411029606610866992663789045

2897:

2.999999920565026263850135706013774728748876651182343698596498346379356957785768253410663842576040180

4181:

2.999999961862834752781672442791007992946037880433519420759746567240956918340030536674617278961425821

5741:

2.999999979772898737251734351932605316389837207280841298087992036661728800133939352032328880045363306

6466:

2.999999984054529384941766616770705143588540588662659149701572624953407266189052452652949367029127980

7561:

2.999999988338611459186282879178748135958486797383190644017503575434657178028779924573743743852742618

9077:

2.999999991908592556854544098127663938729200680122304394844671754625176984651727171132527520688953123

10946:

2.999999994435862499082299842016667096630867703622594763199265196843969466997876761517903274789621835

14701:

2.999999996915285421277783704641364238689615847791052784945888026131867356756214090984043596760709292

28657:

2.999999999188203277427941797472261507646812597185212051717678819548951128573556517641223737862893755

33461:

2.999999999404569730114863040203170627286840742447715962079206517422988211508262200803069337992738497

37666:

2.999999999530095356293426153943496144521454514431048173089549764861141516635643865811829995654247981

43261:

2.999999999643782187253800355567385824227911885938199185845976822257516248472445341343322547506111374

51641:

2.999999999750011832023319725602513337814196386549939549100590922184441437917861584110944018149811793

62210:

2.999999999827738454912179598798992333281149850968665293545420338847423411020985667155271227695177245

75025:

2.999999999881560454336200652665239300674242567267524921048762324233410264415102192327982339221767701

96557:

2.999999999928494209331884567538540567893593315028046545139414940452129907060168277466203431201472617

135137:

2.999999999963494340896254617770874210029830674197163448532161501935072420409018419768159725093307296

195025:

2.999999999982472169378956873247480538344038910442848451141202477886405440357067065624530765524475895

196418:

2.999999999982719903168613488079146633338548850581207035779347395023970456498698058368385560881455313

294685:

2.999999999992322979843593648147339006847023963422457847733140470339039881414347493741287566712391262

426389:

2.999999999996333119503835395890926695040463393318576549614736190375742513190712806594469767612290804

499393:

2.999999999997326846857183796793834309072525776578006578475761634194339252804280423864573942202236128

514229:

2.999999999997478867849269527494696871558409467664829240148968522637745793562439931007181039935292088

646018:

2.999999999998402578581897698072116659634734354663454099022747147727625297178494283173812392530911958

925765:

2.999999999999222129635757983148197735267279136410328205661616269820679636980722115946420382089146588

1136689:

2.999999999999484028841356447669420988604220358551121039384097013347384254884671614052096357975154485

1278818:

2.999999999999592346420750344345733485509980345148942898142834252233482803645900952921267449827818531

1346269:

2.999999999999632171776383376910636153393713597500002067375643271662006649542031000686634371551228758

1441889:

2.999999999999679339781900856935844557855729100752772374676343575289698481993843093728117290636224870

1686049:

2.999999999999765486225315992261473027206933012622690312213471176651005225333664631079714603795355363

2012674:

2.999999999999835425756200790927176019615367896193866817642948608864397346152645168891949768669337173

2423525:

2.999999999999886495329363690981687033983394301728733833595168595333157748461688411333525496178468633

2922509:

2.999999999999921945665288317119521959176674686958676774788548149213288257006512466010980661122939776

3276509:

2.999999999999937900799266553314059661213899614592522575505943316004407694113108589206399006396961462

3524578:

2.999999999999946334585416456038192409625246475659733623227596757245243566351453272854534470072639924

4400489:

2.999999999999965572391035774207004674677174522402026709594084356310888904437340242643732757216694826

6625109:

2.999999999999984811227224466968028353363233472428565592888743664304121271883673773308661849229505197

7453378:

2.999999999999987999414191354668582189480173927840574865473589039560879982495126674027070622234653244

8399329:

2.999999999999990550266618955950357352607446131401534756020870600997636651430972189228460841283932700

9227465:

2.999999999999992170321531865276307235989205057237258783059868913988896048524515208559434518716233254

9647009:

2.999999999999992836531202120192166735424318244499928956810063406787651882973336096508267525110746049

11485154:

2.999999999999994946001656612566547906521398370984147750809329221920218906863540769408001770345612160

13782649:

2.999999999999996490515339227724315676355085193805437333435859609702069428699965249898896813976495310

16609837:

2.999999999999997583548962269922152069190735315061036316492109590723634783078146030416575939472210785

16964653:

2.999999999999997683572142338853581021298405948490459837432262623021876143540289184657344733276596084

20031170:

2.999999999999998338516213909512969309184871418555150877868334082983932860687396366595055277006223772

21531778:

2.999999999999998562032885597046415468141919822801095573914084427184296462514814267067045340973615686

24157817:

2.999999999999998857665306601958558567481634051147625796542086788857785857676673431002992561736425723

38613965:

2.999999999999999552884275483073267325321575030237779018465699541608515121846751532777357406383222298

43484701:

2.999999999999999647437786214282625995285401746113202230613777520419971717284085820241837692910111343

48928105:

2.999999999999999721521321428102800797632654970030274748203857953521567856095537135120303708326164278

63245986:

2.999999999999999833335614348456221497945740418209234223178678426488854189484793722280744001250943923

78442645:

2.999999999999999891656145408149388926543113934677446528992115892306853661846851026825230376882654650

94418953:

2.999999999999999925219152909901808935312115333993838702216097319918884484740212564086170366200719589

111242465:

2.999999999999999946127449931400800281600820032936895056268895922621104588477138130146052755742553841

137295677:

2.999999999999999964633261187280780467584745585489019196337358135379929378216563386705626352822095446

144059117:

2.999999999999999967876175543492543765315719922911984616881086647907984734568134519385246495672136243

165580141:

2.999999999999999975683993837235473386440922055739835593640041582050513063551485179719248545849470869

205272962:

2.999999999999999984178587895736961699798073824076568949590567434592617267891795504156870450894137140

225058681:

2.999999999999999986838141957569706950829386578843782479670905038158111212746201848710653916173779332

253191266:

2.999999999999999989600527752287123871218860221343353532416253385880653556041490538356284948856199188

285018617:

2.999999999999999991793418551873951723364096371849090576227570485406375862949923637179231114298924873

298045301:

2.999999999999999992495112564327382665163926588666731608359068211449903867976762679393548665134755152

321534781:

2.999999999999999993551587302092683897168966392650624189902833562169030512015154211524508508248297664

375981346:

2.999999999999999995283974503976597840590665646742732347029528644927870529513850752560681768552053877

433494437:

2.999999999999999996452342512192102455788746237078186534639622541033983725095375383256522505168110898

447626321:

2.999999999999999996672810708512343153881089233459229927152572733754804253741700264138393876197552820

537169541:

2.999999999999999997689606346283457362811537780606276281875035595948425215797984205226936105659876525

576298801:

2.999999999999999997992695265721773239897973295318518793300938980231361822397576587601070123156269554

647072098:

2.999999999999999998407778828416870806144605263943236745949074061915768422591948599827183911396489359

780291637:

2.999999999999999998905048617136988921559597904357308613112628562499362651282097993156130561533904707

941038565:

2.999999999999999999247175234669127888935643563484920382508942041693206292767553729177965965240670407

981277621:

2.999999999999999999307651186855117447581230382561392169040036807688896779513615486963641106352411194

1134903170:

2.999999999999999999482403748109244022235444884331142737908651128078828364432334789877167169389882592

1311738121:

2.999999999999999999612551074296809424262565437411188571318617123354410340880218923116243288314003811

1405695061:

2.999999999999999999662614430570374857075818189637072631978897461987884903202368725762917571517588365

1475706146:

2.999999999999999999693867830762993956424929147720559164163959089923340046023439676561278721559050528

1873012681:

2.999999999999999999809967752175219683365561544666081035019963159168780433186984559146978352323858330

2151239746:

2.999999999999999999855944078653915329856601343847841878626084376680957798879678201463736675493050864

2561077037:

2.999999999999999999898360280806897518327805505590612656210636958736585623876034105398298330894441843

2971215073:

2.999999999999999999924483724572605704503069872743367086160941458396094647364059193374272412856946838

3057250481:

2.999999999999999999928674192529535055855727794260070186345239075546683223604960359880717339922013046

3778847945:

2.999999999999999999953313651554669063834329010080187649348179132160885276967204961823078983658380397

4434764269:

2.999999999999999999966102500621353314511944542002016336374424694757101172965144855414347441177075411

4801489937:

2.999999999999999999971082769622449099141292507775483121570890620501431349079092128365994287741933872

5348189873:

2.999999999999999999976692531480000031163657086529571437166372447757537564059095354702446380583066498

6449974274:

2.999999999999999999983975194165382867764487927515185469715923060145970707387982618331199930885880946

6684339842:

2.999999999999999999985079214157729234994670912141873679311276893016271474990633068928991368610039846

7645370045:

2.999999999999999999988594568521813509125061299799810049274099614100952708434587024608934294857134924

7778742049:

2.999999999999999999988982323898995909384891161015812173649069401128107131382252936934783272782462364

9629807441:

2.999999999999999999992810916383270204789868599181113803839834540036331976235065488890640492875085995

9676815637:

2.999999999999999999992880593232529598024547787117655539763313234169846594181369282630639323681876943

10910721905:

2.999999999999999999994399822815441171087707339152788162544753535176138918356547088749359324325533968

14001740009:

2.999999999999999999996599484783607715171132522374804000588529280466010079405992323350288649411555170

14653451665:

2.999999999999999999996895234036763791248768738944286365244145115636330029030178232089374404899344287

17445941365:

2.999999999999999999997809617686979621708424305842628007379396520794395967294016916328812419756782466

19577194573:

2.999999999999999999998260566513624187237269740195881629188241504050583494612387370072510156658973909

20365011074:

2.999999999999999999998392542720365661193272333602873165521836077904187122468046428078172036144593581

24793343170:

2.999999999999999999998915477521944139173322595852990447880161502893403376985483083482377577994115309

25209506681:

2.999999999999999999998950988989413514410506769097907481417438811014478709543460875384119750005027993

30395743789:

2.999999999999999999999278422154390012617505989994253687350594197904156918253687395339802320414800232

32124537073:

2.999999999999999999999353996330574970123072476306303638092543957015101296762954615702104406899692482

44208781349:

2.999999999999999999999658892197036304630221661510207569168530032626228176711580555777968084511810941

44560482149:

2.999999999999999999999664255444849886019871658129861469862941954016504111087794113072996546029680853

50130543874:

2.999999999999999999999734720366899952314214651561981663700362941199636773311464378154705650980528605

53316291173:

2.999999999999999999999765475143563718968059962132001375104817871142677209843608669203134635785367199

56399710225:

2.999999999999999999999790417510062284540800062120826625006357861306366474690425034247586799776540027

63557570729:

2.999999999999999999999834965720276001694873044876216434887257100192490667863746008462455005899216174

65995009186:

2.999999999999999999999846931243657172781756005388544756694848702578231670100072220992257438220110448

71700814274:

2.999999999999999999999870323713465218488268489130054546369029340735705674878086340135144768861628180

83842154089:

2.999999999999999999999905161672729288487101476731761562045937509937915080291489301316570502618857908

95969331289:

2.999999999999999999999927615795877842038120803233157655752069931131761804630122957258105318254727762

99816291793:

2.999999999999999999999933087712426111540927482993408067287055931405971967272908123189424006994973538

119154326114:

2.999999999999999999999953044214362883728328983946549239957338042642643698875403339874735614827691699

128367472469:

2.999999999999999999999959542526277800287731187152347749668783844430244359012931894622491769334849499

139583862445:

2.999999999999999999999965783284580371583148354687546758821184283008997954913762203096520322601570997

143367113573:

2.999999999999999999999967565317001754945934984050433611444927197118950107929414712297759237015183003

162930183509:

2.999999999999999999999974886592275330312704077288858291494320350370783675195433175945493999452497150

172765826641:

2.999999999999999999999977664634899821211465483302722817526615865476773197508853770796184228795140583

208333239010:

2.999999999999999999999984639986091449114314366825824433419275193469571015031412517701345277778591956

251250963713:

2.999999999999999999999989439286315256241308231105101981814742000878841581181559776748386245544050317

259717522849:

2.999999999999999999999990116603082615550601378976952947249112182912367438693221367070986143136005879

292182217634:

2.999999999999999999999992190896499546903357337158259688293385742731854042588085785683895712211347193

303011495165:

2.999999999999999999999992739098718418654221130218306947680316386536654476825516327624841986387591090

311809494089:

2.999999999999999999999993143064382462151499072765517254357108625140960654199704724104847368731684395

328716329765:

2.999999999999999999999993830269663972146327379475079161067925939863864170997943309997720929844734055

365435296162:

2.999999999999999999999995007848498882113978540974408581179314599661559685636345533049371253089962365

370238963953:

2.999999999999999999999995136549533766844301583155040546427087992324982956821365197489701871578022583

417673428514:

2.999999999999999999999996178489594789325176103074545579218317750735396742850530150833078259861034764

479716816349:

2.999999999999999999999997103064303007592128686578827800328506133295214342293455450975813380569055878

571101889373:

2.999999999999999999999997955996914428596286229123271931592805013155309710434714773077805897964187670

665048316673:

2.999999999999999999999998492690820987540814558078719467401136546350416180934491490563845473150981450

679944086914:

2.999999999999999999999998558009702223068971625114750067878698684756407360708853563596572562965765264

815959972309:

2.999999999999999999999998998684285786509701938020321590727541840511558180474048910979516849707783520

841771717954:

2.999999999999999999999999059150668454308425188198116229592392075572139539371818716222866655966725597

956722026041:

2.999999999999999999999999271654911803214701432565287304009319669110575560588939331982190879181817753

982145940029:

2.999999999999999999999999308874862221557510906191569712227934999823711825072055993584098929283839936

Definition

For a Markov number $m$ [5], the Lagrange number in the Markov spectrum [4] is $L_m=\sqrt{9m^2-4}/m$. These are the points of the Markov and Lagrange spectra in $[\sqrt5,3)$ [1] [2].

Parameters

$m$

— Markov number ($m$ is the largest member of a normalised Markov triple)

Formulas

(1)

$m_1^2+m_2^2+m^2=3m_1m_2m$ for a Markov triple.

(2)

$L_m=\frac{\sqrt{9m^2-4}}{m}=3\sqrt{1-\frac{4}{9m^2}}$.

(3)

The corresponding Markov form has discriminant $D=9m^2-4$, so $L_m=\sqrt D/m$ [1].

Comments

(4)

Below $3$, the Markov spectrum and the Lagrange spectrum agree and form this discrete set [2]. MathWorld also uses "Lagrange number" for an unrelated Pell-equation family [6]; this table uses the Diophantine approximation meaning.

(5)

The values increase to $3$ from below. Since $3-L_m=4/(m(3m+\sqrt{9m^2-4}))$, later entries agree with $3$ in about $2\log_{10}m$ leading decimal places.

(6)

The entries are ordered by increasing Markov number $m$, which is the order obtained by enumerating the normalised Markov triples by $m$, then by $m_2$, then by $m_1$.

Programs

(P1)

Sage

R = RealBallField(400)
def lagrange_number(m):
    return R(9*m*m - 4).sqrt() / m
[lagrange_number(m) for m in [1, 2, 5, 13]]

References

[1]

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

[2]

T. W. Cusick and M. E. Flahive, The Markoff and Lagrange Spectra, Mathematical Surveys and Monographs 30, American Mathematical Society, 1989.

[3]

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

Links

[4]

Wikipedia: Markov spectrum

[5]

Wikipedia: Markov number

[6]

MathWorld: Lagrange Number

[7]

MathWorld: Markov Number

[8]

MathWorld: Markov Spectrum

[9]

OEIS A002559: Markov numbers

[10]

OEIS A200991: decimal expansion of $L_5$

[11]

OEIS A305308: decimal expansion of $L_{13}$

Similar tables

Hermite's constants — the same extremal problem for positive definite quadratic forms rather than indefinite binary quadratic forms

the golden ratio — has Lagrange number $L_1=\sqrt5$, the first row here

Khinchin's means — almost-sure limits of the regular continued fraction, while the values here describe the extremal approximation cases

Lévy's constant — the almost-sure denominator growth rate for the same regular continued fraction

Lochs's constant — the almost-sure conversion rate between decimal digits and continued-fraction partial quotients

Eigenvalues of the Gauss-Kuzmin-Wirsing operator — the transfer operator for the Gauss continued-fraction map

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds every Markov number $m\leq10^{12}$)

How they were obtained:

The generator traverses the Markov tree from $(1,1,1)$ by Vieta involutions, normalises each triple, and keeps the distinct largest members $m\leq10^{12}$. It computes $\sqrt{9m^2-4}/m$ in real ball arithmetic from the exact integer $9m^2-4$, so each stored decimal is covered by the ball.

The generated Markov numbers were compared with OEIS A002559 [9], and the rows $m=5$ and $m=13$ were compared with OEIS A200991 and A305308 [10] [11].