Markov forms

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Polynomials

$m_1$

$m_2$

$m_3$

$f_{m_1,m_2,m_3}(x,y)$

x^2 + 3*x*y + y^2

2*x^2 + 4*x*y - 2*y^2

5*x^2 + 11*x*y - 5*y^2

13:

13*x^2 + 29*x*y - 13*y^2

34:

34*x^2 + 76*x*y - 34*y^2

89:

89*x^2 + 199*x*y - 89*y^2

233:

233*x^2 + 521*x*y - 233*y^2

233

610:

610*x^2 + 1364*x*y - 610*y^2

610

1597:

1597*x^2 + 3571*x*y - 1597*y^2

1597

4181:

4181*x^2 + 9349*x*y - 4181*y^2

4181

10946:

10946*x^2 + 24476*x*y - 10946*y^2

10946

28657:

28657*x^2 + 64079*x*y - 28657*y^2

28657

75025:

75025*x^2 + 167761*x*y - 75025*y^2

75025

196418:

196418*x^2 + 439204*x*y - 196418*y^2

196418

514229:

514229*x^2 + 1149851*x*y - 514229*y^2

514229

1346269:

1346269*x^2 + 3010349*x*y - 1346269*y^2

1346269

3524578:

3524578*x^2 + 7881196*x*y - 3524578*y^2

3524578

9227465:

9227465*x^2 + 20633239*x*y - 9227465*y^2

9227465

24157817:

24157817*x^2 + 54018521*x*y - 24157817*y^2

24157817

63245986:

63245986*x^2 + 141422324*x*y - 63245986*y^2

63245986

165580141:

165580141*x^2 + 370248451*x*y - 165580141*y^2

165580141

433494437:

433494437*x^2 + 969323029*x*y - 433494437*y^2

433494437

1134903170:

1134903170*x^2 + 2537720636*x*y - 1134903170*y^2

1134903170

2971215073:

2971215073*x^2 + 6643838879*x*y - 2971215073*y^2

2971215073

7778742049:

7778742049*x^2 + 17393796001*x*y - 7778742049*y^2

7778742049

20365011074:

20365011074*x^2 + 45537549124*x*y - 20365011074*y^2

20365011074

53316291173:

53316291173*x^2 + 119218851371*x*y - 53316291173*y^2

53316291173

139583862445:

139583862445*x^2 + 312119004989*x*y - 139583862445*y^2

139583862445

365435296162:

365435296162*x^2 + 817138163596*x*y - 365435296162*y^2

365435296162

956722026041:

956722026041*x^2 + 2139295485799*x*y - 956722026041*y^2

29:

29*x^2 + 63*x*y - 31*y^2

169:

169*x^2 + 367*x*y - 181*y^2

169

985:

985*x^2 + 2139*x*y - 1055*y^2

985

5741:

5741*x^2 + 12467*x*y - 6149*y^2

5741

33461:

33461*x^2 + 72663*x*y - 35839*y^2

33461

195025:

195025*x^2 + 423511*x*y - 208885*y^2

195025

1136689:

1136689*x^2 + 2468403*x*y - 1217471*y^2

1136689

6625109:

6625109*x^2 + 14386907*x*y - 7095941*y^2

6625109

38613965:

38613965*x^2 + 83853039*x*y - 41358175*y^2

38613965

225058681:

225058681*x^2 + 488731327*x*y - 241053109*y^2

225058681

1311738121:

1311738121*x^2 + 2848534923*x*y - 1404960479*y^2

1311738121

7645370045:

7645370045*x^2 + 16602478211*x*y - 8188709765*y^2

7645370045

44560482149:

44560482149*x^2 + 96766334343*x*y - 47727298111*y^2

44560482149

259717522849:

259717522849*x^2 + 563995527847*x*y - 278175078901*y^2

194:

194*x^2 + 432*x*y - 196*y^2

433:

433*x^2 + 941*x*y - 463*y^2

194

2897:

2897*x^2 + 6451*x*y - 2927*y^2

433

6466:

6466*x^2 + 14052*x*y - 6914*y^2

2897

43261:

43261*x^2 + 96333*x*y - 43709*y^2

6466

96557:

96557*x^2 + 209839*x*y - 103247*y^2

43261

646018:

646018*x^2 + 1438544*x*y - 652708*y^2

96557

1441889:

1441889*x^2 + 3133533*x*y - 1541791*y^2

646018

9647009:

9647009*x^2 + 21481827*x*y - 9746911*y^2

1441889

21531778:

21531778*x^2 + 46793156*x*y - 23023618*y^2

9647009

144059117:

144059117*x^2 + 320788861*x*y - 145550957*y^2

21531778

321534781:

321534781*x^2 + 698763807*x*y - 343812479*y^2

144059117

2151239746:

2151239746*x^2 + 4790351088*x*y - 2173517444*y^2

321534781

4801489937:

4801489937*x^2 + 10434663949*x*y - 5134163567*y^2

2151239746

32124537073:

32124537073*x^2 + 71534477459*x*y - 32457210703*y^2

4801489937

71700814274:

71700814274*x^2 + 155821195428*x*y - 76668641026*y^2

32124537073

479716816349:

479716816349*x^2 + 1068226810797*x*y - 484684643101*y^2

1325:

1325*x^2 + 2961*x*y - 1327*y^2

194

7561:

7561*x^2 + 16837*x*y - 7639*y^2

1325

51641:

51641*x^2 + 115403*x*y - 51719*y^2

7561

294685:

294685*x^2 + 656211*x*y - 297725*y^2

51641

2012674:

2012674*x^2 + 4497756*x*y - 2015714*y^2

294685

11485154:

11485154*x^2 + 25575392*x*y - 11603636*y^2

2012674

78442645:

78442645*x^2 + 175297081*x*y - 78561127*y^2

11485154

447626321:

447626321*x^2 + 996784077*x*y - 452244079*y^2

78442645

3057250481:

3057250481*x^2 + 6832088403*x*y - 3061868239*y^2

447626321

17445941365:

17445941365*x^2 + 38849003611*x*y - 17625915445*y^2

3057250481

119154326114:

119154326114*x^2 + 266276150636*x*y - 119334300194*y^2

17445941365

679944086914:

679944086914*x^2 + 1514114356752*x*y - 686958458276*y^2

169

14701:

14701*x^2 + 31925*x*y - 15745*y^2

433

37666:

37666*x^2 + 81856*x*y - 40276*y^2

14701

1278818:

1278818*x^2 + 2777108*x*y - 1369634*y^2

37666

3276509:

3276509*x^2 + 7120531*x*y - 3503549*y^2

1278818

111242465:

111242465*x^2 + 241576471*x*y - 119142413*y^2

3276509

285018617:

285018617*x^2 + 619404341*x*y - 304768487*y^2

111242465

9676815637:

9676815637*x^2 + 21014375869*x*y - 10364020297*y^2

285018617

24793343170:

24793343170*x^2 + 53881057136*x*y - 26511354820*y^2

9676815637

841771717954:

841771717954*x^2 + 1828009124132*x*y - 901550623426*y^2

9077:

9077*x^2 + 20295*x*y - 9079*y^2

1325

135137:

135137*x^2 + 301993*x*y - 135341*y^2

9077

925765:

925765*x^2 + 2069891*x*y - 925969*y^2

135137

13782649:

13782649*x^2 + 30800325*x*y - 13803455*y^2

925765

94418953:

94418953*x^2 + 211108587*x*y - 94439759*y^2

13782649

1405695061:

1405695061*x^2 + 3141331157*x*y - 1407817069*y^2

94418953

9629807441:

9629807441*x^2 + 21531005983*x*y - 9631929449*y^2

1405695061

143367113573:

143367113573*x^2 + 320384977689*x*y - 143583537583*y^2

9629807441

982145940029:

982145940029*x^2 + 2195951501679*x*y - 982362364039*y^2

233

62210:

62210*x^2 + 139104*x*y - 62212*y^2

9077

2423525:

2423525*x^2 + 5418689*x*y - 2424059*y^2

62210

16609837:

16609837*x^2 + 37140247*x*y - 16610371*y^2

2423525

647072098:

647072098*x^2 + 1446769668*x*y - 647214674*y^2

16609837

4434764269:

4434764269*x^2 + 9916306845*x*y - 4434906845*y^2

647072098

172765826641:

172765826641*x^2 + 386282082667*x*y - 172803893899*y^2

169

985

499393:

499393*x^2 + 1084469*x*y - 534883*y^2

169

14701

7453378:

7453378*x^2 + 16185912*x*y - 7982684*y^2

169

499393

253191266:

253191266*x^2 + 549823644*x*y - 271184626*y^2

169

7453378

3778847945:

3778847945*x^2 + 8206225459*x*y - 4047205043*y^2

169

253191266

128367472469:

128367472469*x^2 + 278759503039*x*y - 137490070499*y^2

194

2897

1686049:

1686049*x^2 + 3754471*x*y - 1703509*y^2

194

7561

4400489:

4400489*x^2 + 9799105*x*y - 4445885*y^2

194

1686049

981277621:

981277621*x^2 + 2185095671*x*y - 991439311*y^2

194

4400489

2561077037:

2561077037*x^2 + 5703062273*x*y - 2587497431*y^2

194

981277621

571101889373:

571101889373*x^2 + 1271721926051*x*y - 577015975493*y^2

233

610

426389:

426389*x^2 + 953433*x*y - 426391*y^2

233

62210

43484701:

43484701*x^2 + 97233497*x*y - 43486099*y^2

233

426389

298045301:

298045301*x^2 + 666448303*x*y - 298046699*y^2

233

43484701

30395743789:

30395743789*x^2 + 67966075299*x*y - 30396720989*y^2

233

298045301

208333239010:

208333239010*x^2 + 465846410364*x*y - 208334216210*y^2

433

6466

8399329:

8399329*x^2 + 18253537*x*y - 8981281*y^2

433

37666

48928105:

48928105*x^2 + 106330881*x*y - 52318493*y^2

433

8399329

10910721905:

10910721905*x^2 + 23711330511*x*y - 11666677105*y^2

433

48928105

63557570729:

63557570729*x^2 + 138123732563*x*y - 67961682131*y^2

610

1597

2922509:

2922509*x^2 + 6534927*x*y - 2922511*y^2

610

426389

780291637:

780291637*x^2 + 1744781869*x*y - 780295297*y^2

610

2922509

5348189873:

5348189873*x^2 + 11958912839*x*y - 5348193533*y^2

985

5741

16964653:

16964653*x^2 + 36839981*x*y - 18170293*y^2

985

499393

1475706146:

1475706146*x^2 + 3204605528*x*y - 1580579084*y^2

985

16964653

50130543874:

50130543874*x^2 + 108862131388*x*y - 53693209666*y^2

1325

51641

205272962:

205272962*x^2 + 458726896*x*y - 205583012*y^2

1325

135137

537169541:

537169541*x^2 + 1200422099*x*y - 537980441*y^2

1325

205272962

815959972309:

815959972309*x^2 + 1823439296197*x*y - 817192420981*y^2

1597

4181

20031170:

20031170*x^2 + 44791056*x*y - 20031172*y^2

1597

2922509

14001740009:

14001740009*x^2 + 31308833893*x*y - 14001749591*y^2

1597

20031170

95969331289:

95969331289*x^2 + 214593939947*x*y - 95969340871*y^2

2897

43261

375981346:

375981346*x^2 + 837230092*x*y - 379874914*y^2

2897

1686049

14653451665:

14653451665*x^2 + 32630107029*x*y - 14805196523*y^2

4181

10946

137295677:

137295677*x^2 + 307002465*x*y - 137295679*y^2

4181

20031170

251250963713:

251250963713*x^2 + 561814211837*x*y - 251250988799*y^2

5741

33461

576298801:

576298801*x^2 + 1251474845*x*y - 617255095*y^2

5741

16964653

292182217634:

292182217634*x^2 + 634494990624*x*y - 312946955284*y^2

6466

96557

1873012681:

1873012681*x^2 + 4070456911*x*y - 2002785301*y^2

6466

8399329

162930183509:

162930183509*x^2 + 354082109785*x*y - 174218888375*y^2

7561

294685

6684339842:

6684339842*x^2 + 14884834084*x*y - 6753296162*y^2

7561

4400489

99816291793:

99816291793*x^2 + 222273098283*x*y - 100846009259*y^2

9077

925765

25209506681:

25209506681*x^2 + 56365201745*x*y - 25215061805*y^2

9077

2423525

65995009186:

65995009186*x^2 + 147556319960*x*y - 66009550540*y^2

10946

28657

941038565:

941038565*x^2 + 2104226199*x*y - 941038567*y^2

14701

1278818

56399710225:

56399710225*x^2 + 122478794061*x*y - 60404968271*y^2

14701

7453378

328716329765:

328716329765*x^2 + 713847276569*x*y - 352060312271*y^2

28657

75025

6449974274:

6449974274*x^2 + 14422580928*x*y - 6449974276*y^2

33461

195025

19577194573:

19577194573*x^2 + 42513304709*x*y - 20968502953*y^2

37666

3276509

370238963953:

370238963953*x^2 + 804605761875*x*y - 395894029871*y^2

43261

646018

83842154089:

83842154089*x^2 + 186698555941*x*y - 84710402359*y^2

51641

2012674

311809494089:

311809494089*x^2 + 696805852759*x*y - 312280460009*y^2

75025

196418

44208781349:

44208781349*x^2 + 98853840297*x*y - 44208781351*y^2

96557

1441889

417673428514:

417673428514*x^2 + 907693637632*x*y - 446612140756*y^2

195025

1136689

665048316673:

665048316673*x^2 + 1444200885221*x*y - 712311845323*y^2

196418

514229

303011495165:

303011495165*x^2 + 677554301151*x*y - 303011495167*y^2

Definition

For a normalized Markov triple $(m_1,m_2,m_3)$ [3], this table stores Markov's binary quadratic form $f_{m_1,m_2,m_3}(x,y)$ defined by Formula (1) [1].

Parameters

$m_1$

— smallest member of the triple ($(m_1,m_2,m_3)$ is a Markov triple with $m_1\leq m_2\leq m_3$)

$m_2$

— middle member of the triple

$m_3$

— largest member of the triple

Formulas

(1)

Put $m=m_3$. Let $0\leq k\leq m/2$ be the unique integer satisfying $k m_2\equiv\pm m_1\pmod m$, and let $l=(k^2+1)/m$. Then $f_{m_1,m_2,m_3}(x,y)=m x^2+(3m-2k)xy+(l-3k)y^2$.

(2)

The indexing triples satisfy $m_1^2+m_2^2+m_3^2=3m_1m_2m_3$.

(3)

If $f(x,y)=Ax^2+Bxy+Cy^2$ is the stored form, then $B^2-4AC=9m_3^2-4$.

(4)

For the stored form $f(x,y)=Ax^2+Bxy+Cy^2$, put $\mu(f)=\min\{0<|f(x,y)|:(x,y)\in\mathbb Z^2\}$. The Markov value is $\sqrt{B^2-4AC}/\mu(f)=\sqrt{9m_3^2-4}/m_3$.

(5)

$\gcd(A,B,C)=2$ when $m_3$ is even, and $\gcd(A,B,C)=1$ when $m_3$ is odd.

Comments

(6)

The triples are normalized by sorting them as $m_1\leq m_2\leq m_3$. Rows are grouped by increasing $m_1$ and, within each $m_1$ group, are listed by increasing $m_3$, then by increasing $m_2$. The table is keyed by the full triple rather than by $m_3$ alone, so it does not use the Markov unicity conjecture [2].

(7)

This table stores Markov's form itself, with leading coefficient $m_3$, rather than its primitive part.

(8)

In Formula (1), the sign in $k m_2\equiv\pm m_1\pmod {m_3}$ gives the residues $k$ and $m_3-k$ modulo $m_3$. Markov's theorem [3] states this range strictly, $0<k<m_3/2$; this table takes the range closed so that the degenerate triples $(1,1,1)$ and $(1,1,2)$ have a $k$, giving $k=0$ and $k=1$.

(9)

Writing $f(x,y)=Ax^2+Bxy+Cy^2$, the root $(-B+\sqrt{B^2-4AC})/(2A)$ of $f(x,1)$ has an eventually periodic continued fraction; for every row here, a period uses only the partial quotients $1$ and $2$. The associated value in the Markov and Lagrange spectra [4] is $\sqrt{9m_3^2-4}/m_3=3-2/(3m_3^2)+O(m_3^{-4})$, so these values accumulate at $3$ from smaller values.

Programs

(P1)

Sage

from sage.rings.integer_ring import ZZ
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing

R = PolynomialRing(ZZ, ("x", "y"))
x, y = R.gens()
m1, m2, m3 = ZZ(2), ZZ(5), ZZ(29)
if m3 == 1:
    k = ZZ(0)
else:
    r = (m1 * ZZ(pow(int(m2), -1, int(m3)))) % m3
    k = min(r, (-r) % m3)
l = (k**2 + 1) // m3
m3*x**2 + (3*m3 - 2*k)*x*y + (l - 3*k)*y**2

References

[1]

J. W. S. Cassels, An Introduction to Diophantine Approximation, Cambridge Tracts in Mathematics and Mathematical Physics 45, Cambridge University Press, 1957.

[2]

Martin Aigner, Markov's Theorem and 100 Years of the Uniqueness Conjecture, Springer, 2013.

Links

[3]

Wikipedia: Markov number

[4]

Wikipedia: Markov spectrum

[5]

OEIS A291694: Markov triples ordered by largest entry

[6]

OEIS A305310: the parameter $k$ for Markov triples

Similar tables

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Pisot numbers less than the golden ratio — is another non-rectangular family, but it stores real algebraic numbers rather than binary quadratic forms

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Data properties

Entries are of type: integral polynomial

How well the digits are known: exact

Table is complete: no (it holds every normalized Markov triple with $m_3\leq10^{12}$)

How they were obtained:

The generator enumerates triples from $(1,1,1)$ by the Vieta involutions of Formula (2), sorts each triple into normalized order, and keeps exactly the triples with $m_3\leq10^{12}$. It computes $k$ by modular inversion in the congruence from the definition, using exact integer arithmetic throughout.

For every stored row, the generator verifies the Markov equation, the divisibility of $k^2+1$ by $m_3$, Formula (3), and Formula (5). The first 23 triples were compared with [5], and the first forty $k$ values were compared with [6].