Markov triples

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Numbers

$m_1$, $m_2$, $m_3$

$m_i$

1, 1, 1

$m_1$:

1, 1, 1

$m_2$:

1, 1, 1

$m_3$:

1, 1, 2

$m_1$:

1, 1, 2

$m_2$:

1, 1, 2

$m_3$:

1, 2, 5

$m_1$:

1, 2, 5

$m_2$:

1, 2, 5

$m_3$:

1, 5, 13

$m_1$:

1, 5, 13

$m_2$:

1, 5, 13

$m_3$:

2, 5, 29

$m_1$:

2, 5, 29

$m_2$:

2, 5, 29

$m_3$:

1, 13, 34

$m_1$:

1, 13, 34

$m_2$:

1, 13, 34

$m_3$:

1, 34, 89

$m_1$:

1, 34, 89

$m_2$:

1, 34, 89

$m_3$:

2, 29, 169

$m_1$:

2, 29, 169

$m_2$:

2, 29, 169

$m_3$:

169

5, 13, 194

$m_1$:

5, 13, 194

$m_2$:

5, 13, 194

$m_3$:

194

1, 89, 233

$m_1$:

1, 89, 233

$m_2$:

1, 89, 233

$m_3$:

233

5, 29, 433

$m_1$:

5, 29, 433

$m_2$:

5, 29, 433

$m_3$:

433

1, 233, 610

$m_1$:

1, 233, 610

$m_2$:

233

1, 233, 610

$m_3$:

610

2, 169, 985

$m_1$:

2, 169, 985

$m_2$:

169

2, 169, 985

$m_3$:

985

13, 34, 1325

$m_1$:

13, 34, 1325

$m_2$:

13, 34, 1325

$m_3$:

1325

1, 610, 1597

$m_1$:

1, 610, 1597

$m_2$:

610

1, 610, 1597

$m_3$:

1597

5, 194, 2897

$m_1$:

5, 194, 2897

$m_2$:

194

5, 194, 2897

$m_3$:

2897

1, 1597, 4181

$m_1$:

1, 1597, 4181

$m_2$:

1597

1, 1597, 4181

$m_3$:

4181

2, 985, 5741

$m_1$:

2, 985, 5741

$m_2$:

985

2, 985, 5741

$m_3$:

5741

5, 433, 6466

$m_1$:

5, 433, 6466

$m_2$:

433

5, 433, 6466

$m_3$:

6466

13, 194, 7561

$m_1$:

13, 194, 7561

$m_2$:

194

13, 194, 7561

$m_3$:

7561

34, 89, 9077

$m_1$:

34, 89, 9077

$m_2$:

34, 89, 9077

$m_3$:

9077

1, 4181, 10946

$m_1$:

1, 4181, 10946

$m_2$:

4181

1, 4181, 10946

$m_3$:

10946

29, 169, 14701

$m_1$:

29, 169, 14701

$m_2$:

169

29, 169, 14701

$m_3$:

14701

1, 10946, 28657

$m_1$:

1, 10946, 28657

$m_2$:

10946

1, 10946, 28657

$m_3$:

28657

2, 5741, 33461

$m_1$:

2, 5741, 33461

$m_2$:

5741

2, 5741, 33461

$m_3$:

33461

29, 433, 37666

$m_1$:

29, 433, 37666

$m_2$:

433

29, 433, 37666

$m_3$:

37666

5, 2897, 43261

$m_1$:

5, 2897, 43261

$m_2$:

2897

5, 2897, 43261

$m_3$:

43261

13, 1325, 51641

$m_1$:

13, 1325, 51641

$m_2$:

1325

13, 1325, 51641

$m_3$:

51641

89, 233, 62210

$m_1$:

89, 233, 62210

$m_2$:

233

89, 233, 62210

$m_3$:

62210

1, 28657, 75025

$m_1$:

1, 28657, 75025

$m_2$:

28657

1, 28657, 75025

$m_3$:

75025

5, 6466, 96557

$m_1$:

5, 6466, 96557

$m_2$:

6466

5, 6466, 96557

$m_3$:

96557

34, 1325, 135137

$m_1$:

34, 1325, 135137

$m_2$:

1325

34, 1325, 135137

$m_3$:

135137

2, 33461, 195025

$m_1$:

2, 33461, 195025

$m_2$:

33461

2, 33461, 195025

$m_3$:

195025

1, 75025, 196418

$m_1$:

1, 75025, 196418

$m_2$:

75025

1, 75025, 196418

$m_3$:

196418

13, 7561, 294685

$m_1$:

13, 7561, 294685

$m_2$:

7561

13, 7561, 294685

$m_3$:

294685

233, 610, 426389

$m_1$:

233

233, 610, 426389

$m_2$:

610

233, 610, 426389

$m_3$:

426389

169, 985, 499393

$m_1$:

169

169, 985, 499393

$m_2$:

985

169, 985, 499393

$m_3$:

499393

1, 196418, 514229

$m_1$:

1, 196418, 514229

$m_2$:

196418

1, 196418, 514229

$m_3$:

514229

5, 43261, 646018

$m_1$:

5, 43261, 646018

$m_2$:

43261

5, 43261, 646018

$m_3$:

646018

34, 9077, 925765

$m_1$:

34, 9077, 925765

$m_2$:

9077

34, 9077, 925765

$m_3$:

925765

2, 195025, 1136689

$m_1$:

2, 195025, 1136689

$m_2$:

195025

2, 195025, 1136689

$m_3$:

1136689

29, 14701, 1278818

$m_1$:

29, 14701, 1278818

$m_2$:

14701

29, 14701, 1278818

$m_3$:

1278818

1, 514229, 1346269

$m_1$:

1, 514229, 1346269

$m_2$:

514229

1, 514229, 1346269

$m_3$:

1346269

5, 96557, 1441889

$m_1$:

5, 96557, 1441889

$m_2$:

96557

5, 96557, 1441889

$m_3$:

1441889

194, 2897, 1686049

$m_1$:

194

194, 2897, 1686049

$m_2$:

2897

194, 2897, 1686049

$m_3$:

1686049

13, 51641, 2012674

$m_1$:

13, 51641, 2012674

$m_2$:

51641

13, 51641, 2012674

$m_3$:

2012674

89, 9077, 2423525

$m_1$:

89, 9077, 2423525

$m_2$:

9077

89, 9077, 2423525

$m_3$:

2423525

610, 1597, 2922509

$m_1$:

610

610, 1597, 2922509

$m_2$:

1597

610, 1597, 2922509

$m_3$:

2922509

29, 37666, 3276509

$m_1$:

29, 37666, 3276509

$m_2$:

37666

29, 37666, 3276509

$m_3$:

3276509

1, 1346269, 3524578

$m_1$:

1, 1346269, 3524578

$m_2$:

1346269

1, 1346269, 3524578

$m_3$:

3524578

194, 7561, 4400489

$m_1$:

194

194, 7561, 4400489

$m_2$:

7561

194, 7561, 4400489

$m_3$:

4400489

2, 1136689, 6625109

$m_1$:

2, 1136689, 6625109

$m_2$:

1136689

2, 1136689, 6625109

$m_3$:

6625109

169, 14701, 7453378

$m_1$:

169

169, 14701, 7453378

$m_2$:

14701

169, 14701, 7453378

$m_3$:

7453378

433, 6466, 8399329

$m_1$:

433

433, 6466, 8399329

$m_2$:

6466

433, 6466, 8399329

$m_3$:

8399329

1, 3524578, 9227465

$m_1$:

1, 3524578, 9227465

$m_2$:

3524578

1, 3524578, 9227465

$m_3$:

9227465

5, 646018, 9647009

$m_1$:

5, 646018, 9647009

$m_2$:

646018

5, 646018, 9647009

$m_3$:

9647009

13, 294685, 11485154

$m_1$:

13, 294685, 11485154

$m_2$:

294685

13, 294685, 11485154

$m_3$:

11485154

34, 135137, 13782649

$m_1$:

34, 135137, 13782649

$m_2$:

135137

34, 135137, 13782649

$m_3$:

13782649

89, 62210, 16609837

$m_1$:

89, 62210, 16609837

$m_2$:

62210

89, 62210, 16609837

$m_3$:

16609837

985, 5741, 16964653

$m_1$:

985

985, 5741, 16964653

$m_2$:

5741

985, 5741, 16964653

$m_3$:

16964653

1597, 4181, 20031170

$m_1$:

1597

1597, 4181, 20031170

$m_2$:

4181

1597, 4181, 20031170

$m_3$:

20031170

5, 1441889, 21531778

$m_1$:

5, 1441889, 21531778

$m_2$:

1441889

5, 1441889, 21531778

$m_3$:

21531778

1, 9227465, 24157817

$m_1$:

1, 9227465, 24157817

$m_2$:

9227465

1, 9227465, 24157817

$m_3$:

24157817

2, 6625109, 38613965

$m_1$:

2, 6625109, 38613965

$m_2$:

6625109

2, 6625109, 38613965

$m_3$:

38613965

233, 62210, 43484701

$m_1$:

233

233, 62210, 43484701

$m_2$:

62210

233, 62210, 43484701

$m_3$:

43484701

433, 37666, 48928105

$m_1$:

433

433, 37666, 48928105

$m_2$:

37666

433, 37666, 48928105

$m_3$:

48928105

1, 24157817, 63245986

$m_1$:

1, 24157817, 63245986

$m_2$:

24157817

1, 24157817, 63245986

$m_3$:

63245986

13, 2012674, 78442645

$m_1$:

13, 2012674, 78442645

$m_2$:

2012674

13, 2012674, 78442645

$m_3$:

78442645

34, 925765, 94418953

$m_1$:

34, 925765, 94418953

$m_2$:

925765

34, 925765, 94418953

$m_3$:

94418953

29, 1278818, 111242465

$m_1$:

29, 1278818, 111242465

$m_2$:

1278818

29, 1278818, 111242465

$m_3$:

111242465

4181, 10946, 137295677

$m_1$:

4181

4181, 10946, 137295677

$m_2$:

10946

4181, 10946, 137295677

$m_3$:

137295677

5, 9647009, 144059117

$m_1$:

5, 9647009, 144059117

$m_2$:

9647009

5, 9647009, 144059117

$m_3$:

144059117

1, 63245986, 165580141

$m_1$:

1, 63245986, 165580141

$m_2$:

63245986

1, 63245986, 165580141

$m_3$:

165580141

1325, 51641, 205272962

$m_1$:

1325

1325, 51641, 205272962

$m_2$:

51641

1325, 51641, 205272962

$m_3$:

205272962

2, 38613965, 225058681

$m_1$:

2, 38613965, 225058681

$m_2$:

38613965

2, 38613965, 225058681

$m_3$:

225058681

169, 499393, 253191266

$m_1$:

169

169, 499393, 253191266

$m_2$:

499393

169, 499393, 253191266

$m_3$:

253191266

29, 3276509, 285018617

$m_1$:

29, 3276509, 285018617

$m_2$:

3276509

29, 3276509, 285018617

$m_3$:

285018617

233, 426389, 298045301

$m_1$:

233

233, 426389, 298045301

$m_2$:

426389

233, 426389, 298045301

$m_3$:

298045301

5, 21531778, 321534781

$m_1$:

5, 21531778, 321534781

$m_2$:

21531778

5, 21531778, 321534781

$m_3$:

321534781

2897, 43261, 375981346

$m_1$:

2897

2897, 43261, 375981346

$m_2$:

43261

2897, 43261, 375981346

$m_3$:

375981346

1, 165580141, 433494437

$m_1$:

1, 165580141, 433494437

$m_2$:

165580141

1, 165580141, 433494437

$m_3$:

433494437

13, 11485154, 447626321

$m_1$:

13, 11485154, 447626321

$m_2$:

11485154

13, 11485154, 447626321

$m_3$:

447626321

1325, 135137, 537169541

$m_1$:

1325

1325, 135137, 537169541

$m_2$:

135137

1325, 135137, 537169541

$m_3$:

537169541

5741, 33461, 576298801

$m_1$:

5741

5741, 33461, 576298801

$m_2$:

33461

5741, 33461, 576298801

$m_3$:

576298801

89, 2423525, 647072098

$m_1$:

89, 2423525, 647072098

$m_2$:

2423525

89, 2423525, 647072098

$m_3$:

647072098

610, 426389, 780291637

$m_1$:

610

610, 426389, 780291637

$m_2$:

426389

610, 426389, 780291637

$m_3$:

780291637

10946, 28657, 941038565

$m_1$:

10946

10946, 28657, 941038565

$m_2$:

28657

10946, 28657, 941038565

$m_3$:

941038565

194, 1686049, 981277621

$m_1$:

194

194, 1686049, 981277621

$m_2$:

1686049

194, 1686049, 981277621

$m_3$:

981277621

1, 433494437, 1134903170

$m_1$:

1, 433494437, 1134903170

$m_2$:

433494437

1, 433494437, 1134903170

$m_3$:

1134903170

2, 225058681, 1311738121

$m_1$:

2, 225058681, 1311738121

$m_2$:

225058681

2, 225058681, 1311738121

$m_3$:

1311738121

34, 13782649, 1405695061

$m_1$:

34, 13782649, 1405695061

$m_2$:

13782649

34, 13782649, 1405695061

$m_3$:

1405695061

985, 499393, 1475706146

$m_1$:

985

985, 499393, 1475706146

$m_2$:

499393

985, 499393, 1475706146

$m_3$:

1475706146

6466, 96557, 1873012681

$m_1$:

6466

6466, 96557, 1873012681

$m_2$:

96557

6466, 96557, 1873012681

$m_3$:

1873012681

5, 144059117, 2151239746

$m_1$:

5, 144059117, 2151239746

$m_2$:

144059117

5, 144059117, 2151239746

$m_3$:

2151239746

194, 4400489, 2561077037

$m_1$:

194

194, 4400489, 2561077037

$m_2$:

4400489

194, 4400489, 2561077037

$m_3$:

2561077037

1, 1134903170, 2971215073

$m_1$:

1, 1134903170, 2971215073

$m_2$:

1134903170

1, 1134903170, 2971215073

$m_3$:

2971215073

13, 78442645, 3057250481

$m_1$:

13, 78442645, 3057250481

$m_2$:

78442645

13, 78442645, 3057250481

$m_3$:

3057250481

169, 7453378, 3778847945

$m_1$:

169

169, 7453378, 3778847945

$m_2$:

7453378

169, 7453378, 3778847945

$m_3$:

3778847945

89, 16609837, 4434764269

$m_1$:

89, 16609837, 4434764269

$m_2$:

16609837

89, 16609837, 4434764269

$m_3$:

4434764269

5, 321534781, 4801489937

$m_1$:

5, 321534781, 4801489937

$m_2$:

321534781

5, 321534781, 4801489937

$m_3$:

4801489937

610, 2922509, 5348189873

$m_1$:

610

610, 2922509, 5348189873

$m_2$:

2922509

610, 2922509, 5348189873

$m_3$:

5348189873

28657, 75025, 6449974274

$m_1$:

28657

28657, 75025, 6449974274

$m_2$:

75025

28657, 75025, 6449974274

$m_3$:

6449974274

7561, 294685, 6684339842

$m_1$:

7561

7561, 294685, 6684339842

$m_2$:

294685

7561, 294685, 6684339842

$m_3$:

6684339842

2, 1311738121, 7645370045

$m_1$:

2, 1311738121, 7645370045

$m_2$:

1311738121

2, 1311738121, 7645370045

$m_3$:

7645370045

1, 2971215073, 7778742049

$m_1$:

1, 2971215073, 7778742049

$m_2$:

2971215073

1, 2971215073, 7778742049

$m_3$:

7778742049

34, 94418953, 9629807441

$m_1$:

34, 94418953, 9629807441

$m_2$:

94418953

34, 94418953, 9629807441

$m_3$:

9629807441

29, 111242465, 9676815637

$m_1$:

29, 111242465, 9676815637

$m_2$:

111242465

29, 111242465, 9676815637

$m_3$:

9676815637

433, 8399329, 10910721905

$m_1$:

433

433, 8399329, 10910721905

$m_2$:

8399329

433, 8399329, 10910721905

$m_3$:

10910721905

1597, 2922509, 14001740009

$m_1$:

1597

1597, 2922509, 14001740009

$m_2$:

2922509

1597, 2922509, 14001740009

$m_3$:

14001740009

2897, 1686049, 14653451665

$m_1$:

2897

2897, 1686049, 14653451665

$m_2$:

1686049

2897, 1686049, 14653451665

$m_3$:

14653451665

13, 447626321, 17445941365

$m_1$:

13, 447626321, 17445941365

$m_2$:

447626321

13, 447626321, 17445941365

$m_3$:

17445941365

33461, 195025, 19577194573

$m_1$:

33461

33461, 195025, 19577194573

$m_2$:

195025

33461, 195025, 19577194573

$m_3$:

19577194573

1, 7778742049, 20365011074

$m_1$:

1, 7778742049, 20365011074

$m_2$:

7778742049

1, 7778742049, 20365011074

$m_3$:

20365011074

29, 285018617, 24793343170

$m_1$:

29, 285018617, 24793343170

$m_2$:

285018617

29, 285018617, 24793343170

$m_3$:

24793343170

9077, 925765, 25209506681

$m_1$:

9077

9077, 925765, 25209506681

$m_2$:

925765

9077, 925765, 25209506681

$m_3$:

25209506681

233, 43484701, 30395743789

$m_1$:

233

233, 43484701, 30395743789

$m_2$:

43484701

233, 43484701, 30395743789

$m_3$:

30395743789

5, 2151239746, 32124537073

$m_1$:

5, 2151239746, 32124537073

$m_2$:

2151239746

5, 2151239746, 32124537073

$m_3$:

32124537073

75025, 196418, 44208781349

$m_1$:

75025

75025, 196418, 44208781349

$m_2$:

196418

75025, 196418, 44208781349

$m_3$:

44208781349

2, 7645370045, 44560482149

$m_1$:

2, 7645370045, 44560482149

$m_2$:

7645370045

2, 7645370045, 44560482149

$m_3$:

44560482149

985, 16964653, 50130543874

$m_1$:

985

985, 16964653, 50130543874

$m_2$:

16964653

985, 16964653, 50130543874

$m_3$:

50130543874

1, 20365011074, 53316291173

$m_1$:

1, 20365011074, 53316291173

$m_2$:

20365011074

1, 20365011074, 53316291173

$m_3$:

53316291173

14701, 1278818, 56399710225

$m_1$:

14701

14701, 1278818, 56399710225

$m_2$:

1278818

14701, 1278818, 56399710225

$m_3$:

56399710225

433, 48928105, 63557570729

$m_1$:

433

433, 48928105, 63557570729

$m_2$:

48928105

433, 48928105, 63557570729

$m_3$:

63557570729

9077, 2423525, 65995009186

$m_1$:

9077

9077, 2423525, 65995009186

$m_2$:

2423525

9077, 2423525, 65995009186

$m_3$:

65995009186

5, 4801489937, 71700814274

$m_1$:

5, 4801489937, 71700814274

$m_2$:

4801489937

5, 4801489937, 71700814274

$m_3$:

71700814274

43261, 646018, 83842154089

$m_1$:

43261

43261, 646018, 83842154089

$m_2$:

646018

43261, 646018, 83842154089

$m_3$:

83842154089

1597, 20031170, 95969331289

$m_1$:

1597

1597, 20031170, 95969331289

$m_2$:

20031170

1597, 20031170, 95969331289

$m_3$:

95969331289

7561, 4400489, 99816291793

$m_1$:

7561

7561, 4400489, 99816291793

$m_2$:

4400489

7561, 4400489, 99816291793

$m_3$:

99816291793

13, 3057250481, 119154326114

$m_1$:

13, 3057250481, 119154326114

$m_2$:

3057250481

13, 3057250481, 119154326114

$m_3$:

119154326114

169, 253191266, 128367472469

$m_1$:

169

169, 253191266, 128367472469

$m_2$:

253191266

169, 253191266, 128367472469

$m_3$:

128367472469

1, 53316291173, 139583862445

$m_1$:

1, 53316291173, 139583862445

$m_2$:

53316291173

1, 53316291173, 139583862445

$m_3$:

139583862445

34, 1405695061, 143367113573

$m_1$:

34, 1405695061, 143367113573

$m_2$:

1405695061

34, 1405695061, 143367113573

$m_3$:

143367113573

6466, 8399329, 162930183509

$m_1$:

6466

6466, 8399329, 162930183509

$m_2$:

8399329

6466, 8399329, 162930183509

$m_3$:

162930183509

89, 647072098, 172765826641

$m_1$:

89, 647072098, 172765826641

$m_2$:

647072098

89, 647072098, 172765826641

$m_3$:

172765826641

233, 298045301, 208333239010

$m_1$:

233

233, 298045301, 208333239010

$m_2$:

298045301

233, 298045301, 208333239010

$m_3$:

208333239010

4181, 20031170, 251250963713

$m_1$:

4181

4181, 20031170, 251250963713

$m_2$:

20031170

4181, 20031170, 251250963713

$m_3$:

251250963713

2, 44560482149, 259717522849

$m_1$:

2, 44560482149, 259717522849

$m_2$:

44560482149

2, 44560482149, 259717522849

$m_3$:

259717522849

5741, 16964653, 292182217634

$m_1$:

5741

5741, 16964653, 292182217634

$m_2$:

16964653

5741, 16964653, 292182217634

$m_3$:

292182217634

196418, 514229, 303011495165

$m_1$:

196418

196418, 514229, 303011495165

$m_2$:

514229

196418, 514229, 303011495165

$m_3$:

303011495165

51641, 2012674, 311809494089

$m_1$:

51641

51641, 2012674, 311809494089

$m_2$:

2012674

51641, 2012674, 311809494089

$m_3$:

311809494089

14701, 7453378, 328716329765

$m_1$:

14701

14701, 7453378, 328716329765

$m_2$:

7453378

14701, 7453378, 328716329765

$m_3$:

328716329765

1, 139583862445, 365435296162

$m_1$:

1, 139583862445, 365435296162

$m_2$:

139583862445

1, 139583862445, 365435296162

$m_3$:

365435296162

37666, 3276509, 370238963953

$m_1$:

37666

37666, 3276509, 370238963953

$m_2$:

3276509

37666, 3276509, 370238963953

$m_3$:

370238963953

96557, 1441889, 417673428514

$m_1$:

96557

96557, 1441889, 417673428514

$m_2$:

1441889

96557, 1441889, 417673428514

$m_3$:

417673428514

5, 32124537073, 479716816349

$m_1$:

5, 32124537073, 479716816349

$m_2$:

32124537073

5, 32124537073, 479716816349

$m_3$:

479716816349

194, 981277621, 571101889373

$m_1$:

194

194, 981277621, 571101889373

$m_2$:

981277621

194, 981277621, 571101889373

$m_3$:

571101889373

195025, 1136689, 665048316673

$m_1$:

195025

195025, 1136689, 665048316673

$m_2$:

1136689

195025, 1136689, 665048316673

$m_3$:

665048316673

13, 17445941365, 679944086914

$m_1$:

13, 17445941365, 679944086914

$m_2$:

17445941365

13, 17445941365, 679944086914

$m_3$:

679944086914

1325, 205272962, 815959972309

$m_1$:

1325

1325, 205272962, 815959972309

$m_2$:

205272962

1325, 205272962, 815959972309

$m_3$:

815959972309

29, 9676815637, 841771717954

$m_1$:

29, 9676815637, 841771717954

$m_2$:

9676815637

29, 9676815637, 841771717954

$m_3$:

841771717954

1, 365435296162, 956722026041

$m_1$:

1, 365435296162, 956722026041

$m_2$:

365435296162

1, 365435296162, 956722026041

$m_3$:

956722026041

34, 9629807441, 982145940029

$m_1$:

34, 9629807441, 982145940029

$m_2$:

9629807441

34, 9629807441, 982145940029

$m_3$:

982145940029

Definition

A Markov triple is a positive integer solution $(m_1,m_2,m_3)$ of $m_1^2+m_2^2+m_3^2=3m_1m_2m_3$ [3]. This table stores the three components of each normalised triple with $m_1\leq m_2\leq m_3$.

Parameters

$m_1$

— smallest member of the Markov triple ($m_1\geq1$)

$m_2$

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

$m_3$

— largest member of the Markov triple ($m_2\leq m_3$ and $m_1^2+m_2^2+m_3^2=3m_1m_2m_3$)

Formulas

(1)

For a Markov triple $(m_1,m_2,m_3)$, replacing $m_3$ by $3m_1m_2-m_3$, the other root of the Markov equation viewed as a quadratic in $m_3$, gives another Markov triple; the same construction applies to $m_1$ and $m_2$.

(2)

The largest member $m_3$ of a Markov triple gives the Lagrange number $L_{m_3}=\sqrt{9-4/m_3^2}$ in the common part of the Markov and Lagrange spectra below $3$; by Markov's theorem these are all points of the Markov and Lagrange spectra below $3$ [1].

Comments

(3)

The triples are ordered by increasing $m_3$, then by increasing $m_2$, then by increasing $m_1$.

(4)

The table is keyed by the full normalised triple rather than by $m_3$ alone, so it does not use the Markov unicity conjecture [1].

(5)

The spelling Markoff is common in older references and in some OEIS entries.

Programs

(P1)

Python

import heapq

def normalise(triple):
    return tuple(sorted(triple))

def markov_triples(bound):
    start = (1, 1, 1)
    seen = {start}
    heap = [(1, 1, 1, start)]
    while heap:
        _m3, _m2, _m1, triple = heapq.heappop(heap)
        yield triple
        a, b, c = triple
        for nxt in (normalise((3*b*c-a, b, c)),
                    normalise((a, 3*a*c-b, c)),
                    normalise((a, b, 3*a*b-c))):
            if nxt not in seen and nxt[2] <= bound:
                seen.add(nxt)
                heapq.heappush(heap, (nxt[2], nxt[1], nxt[0], nxt))

print(list(markov_triples(1000)))

References

[1]

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

[2]

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

Links

[3]

Wikipedia: Markov number

[4]

Wikipedia: Markov spectrum

[5]

MathWorld: Markov Number

[6]

OEIS A291694: Markov triples ordered by largest entry

[7]

OEIS A002559: Markov numbers

[8]

OEIS A305313: smaller member of the Markov triple

[9]

OEIS A305314: middle member of the Markov triple

Similar tables

Lagrange numbers $L_m$ of the Markov spectrum — stores the spectrum value indexed by the largest member $m_3$ of a Markov triple

Markov forms — stores the binary quadratic forms attached to the same normalised Markov triples

Markov quadratic irrationals — stores the quadratic irrationals attached to the same normalised Markov triples

Hermite's constants — records the analogous extremal constants for positive definite quadratic forms

Khinchin's means — records typical continued-fraction growth rather than the extremal Markov cases

Eigenvalues of the Gauss-Kuzmin-Wirsing operator — concerns the Gauss map behind regular continued fractions

Data properties

Entries are of type: integer

How well the digits are known: exact

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

How they were obtained:

The generator enumerates triples from $(1,1,1)$ by the Vieta involutions of Formula (1), sorts each triple into normalised order, and keeps exactly the triples with $m_3\leq10^{12}$. It verifies the Markov equation on every stored triple and writes the components as exact integers.

The first 40 largest Markov numbers were compared with OEIS A002559 [7]; the corresponding smaller and middle members were compared with OEIS A305313 and A305314 [8] [9]; and the first triples were compared with OEIS A291694 [6].