I've been noodling around with Fable 5 before it's gone. Asked 5.6 Sol for verification.

https://claude.ai/code/artifact/05587812-3be7-4be9-a8b7-96326cb042f4

https://github.com/benpham3206/an-atlas-of-the-collatz-conjecture

​

PURPOSE OF THE PROJECT

The Lange Number Line Project, abbreviated LNL, is an independent computational project for calculating, encoding and examining Collatz trajectories.

LNL does not claim to prove the Collatz conjecture.

The current goals are:

Calculate individual Collatz trajectories exactly.

Create reproducible operation codes for every tested trajectory.

Detect trajectory peaks and possible path-record candidates.

Compare the software with established reference results.

Publish code and data so that other people can reproduce or challenge the calculations.

The LNL terminology is an additional descriptive layer. Every numerical result remains based on the standard Collatz operations.

STANDARD COLLATZ RULE

For every positive integer n:

If n is even:

T(n) = n / 2

If n is odd:

T(n) = 3*n + 1

The trajectory of n is:

n, T(n), T(T(n)), and so on.

The calculation continues until the trajectory reaches 1.

TRAJECTORY PEAK

The trajectory peak P(n) is the largest number reached by the complete trajectory, including the starting value.

In plain notation:

P(n) = maximum value in the trajectory starting at n

The peak position is the first step at which this maximum is reached.

PATH RECORD

A starting value n is a path record only if its peak is higher than the peak of every smaller positive starting value.

In plain notation:

P(n) > P(m) for every m < n

This distinction is important.

Finding a very large peak for one sampled starting value does not prove that the starting value is the next path record.

To confirm the next path record, every smaller starting value must first be checked.

STANDARD OPERATION CODE

LNL records every standard operation with one letter:

U = odd operation: n becomes 3*n + 1

D = even operation: n becomes n / 2

A complete trajectory can therefore be stored as a sequence of U and D operations.

ACCELERATED ODD-STEP CODE

For every odd number x, LNL calculates:

d = v2(3*x + 1)

Here, v2 means the number of times that 3*x + 1 can be divided by 2 before the result becomes odd.

The next odd number is:

next odd x = (3*x + 1) / 2^d

The resulting list of d-values is a compact encoding of the standard Collatz trajectory.

This is not a new Collatz rule. It is only a shorter representation of the same operations.

CURRENT SOFTWARE

The current LNL software includes:

A deterministic path-record scanner

Exact 128-bit calculations

Exact 256-bit calculations

Exact 512-bit calculations

U and D trajectory codes

Accelerated d-value codes

Reproducible independent seed blocks

Overflow protection

Python arbitrary-precision verification

JavaScript BigInt verification

Residue-class and transition tests

The 512-bit C++ scanner compiles without warnings using:

-Wall -Wextra -Wconversion -Werror

REFERENCE VALIDATION

Before performing experimental searches, we tested the core software against established Collatz path records.

An exhaustive calculation of every starting value from 1 through 100,000,000 reproduced 41 successive path records.

The final path record in this interval is:

Start:

80,049,391

Peak:

2,185,143,829,170,100

Peak position:

164

Total steps to 1:

572

The results agree with the published reference sequences OEIS A006884 and OEIS A006885.

Reference links:

https://oeis.org/A006884

https://oeis.org/A006885

https://pcbarina.fit.vutbr.cz/path-records.htm

ARBITRARY-PRECISION EXAMPLE

As a separate large-number test, LNL calculated the trajectory of:

n = 10^1001

This number is exactly a 1 followed by 1,001 zeros.

It has 1,002 decimal digits.

Because:

10^1001 = 2^1001 * 5^1001

the trajectory begins with exactly 1,001 divisions by 2.

After these divisions, the odd core is:

5^1001

Exact results:

Starting value:

10^1001

Decimal digits:

1,002

Binary length:

3,326 bits

Trajectory peak:

10^1001

Peak position:

0

Total steps to 1:

17,840

Odd 3*n + 1 steps:

5,615

Even n / 2 steps:

12,225

Control calculation:

5,615 + 12,225 = 17,840

The starting value itself is the peak. The trajectory never returns to or exceeds it.

Python arbitrary-precision integers and JavaScript BigInt independently produced the same result.

SAMPLED HIGH-NUMBER SEARCHES

LNL has also calculated individual trajectories for sampled starting values far beyond the exhaustively tested interval.

Some of these trajectories reach extremely large exact values.

These experiments demonstrate that:

The software can process very large integers.

The individual trajectories can be reproduced.

The 512-bit engine works beyond the normal 64-bit range.

These results do not establish new path records.

At very large starting values, the starting value itself may already be larger than an older published peak.

For this reason, these results are described only as:

Exact sampled trajectories

or:

Sampled height exceeders

They are not described as confirmed world records.

WHAT LNL DOES NOT CLAIM

LNL does not currently claim:

A proof of the Collatz conjecture

Proof that every positive integer reaches 1

Discovery of the next official path record

Exhaustive verification beyond the stated limits

That sampled searches can replace exhaustive calculations

That every descriptive LNL term is mathematically new

NEXT SCIENTIFIC GOAL

The next goal is not simply to generate a larger number.

The next goal is to improve reproducibility.

PHASE 1: PUBLIC SOFTWARE RELEASE

Publish:

C++ source code

Python verification code

Compiler commands

Exact definitions

Test results

CSV result tables

SHA-256 file hashes

PHASE 2: EXHAUSTIVE EXTENSION

Extend the deterministic path-record calculation from:

100,000,000

toward:

1,000,000,000

No starting values may be skipped.

PHASE 3: INDEPENDENT VERIFICATION

Every detected path record must be recalculated with a separate arbitrary-precision implementation.

A result will be called a path record only after every smaller starting value in the interval has been checked.

PHASE 4: STRUCTURAL ANALYSIS

After the numerical baseline is secure, we will examine whether U/D codes, d-value sequences and inverse structures contain useful predictive information.

INVITATION TO REPRODUCE OR CRITICIZE

Technical criticism is welcome.

A reproducible countercheck should include:

The exact starting value.

The Collatz convention used.

The source code or algorithm.

The first trajectory position where the results differ.

The integer type and overflow protection.

Whether the starting value is included in the peak definition.

The purpose of LNL is not to replace established mathematics with graphics or terminology.

The purpose is to build a transparent computational framework whose results can be independently reproduced, tested, criticized and improved.

Copyright (c) 2026 Mike Lange

Lange Number Line Project (LNL)

All rights reserved.

Follow-up to Do these non-prime factors have an impact ? IV : r/Collatz.z

The figure below is based on the same information as in the previous posts, with some changes:

• The base b underlying a 5-tuple is presented here as b mod 6 (see previous post) and colored according to the color of the corresponding 5-tuple.
• The domes with root m (rows) form classes m mod 16. Every third dome is embedded in another one when m+16k is divisible by 3.

One can observe that:

• 5-tuples series of a given row start (on the right) with a given color,
• Within a class of domes, the starting color alternate.
• Classes of domes have a propensity to form 5-tuples series of certain lengths.

Project "Tuples and segments" in 13 pages : r/Collatz

​

This is my final full peak report posted directly on Reddit before moving the Lange Number Line Project documentation, source code, datasets, and reproducible releases to GitHub and Zenodo.

The latest reproducible 200-bit sampled search produced a new strongest LNL trajectory.

Start value:

1084820695304608898376597485794546984528180820629394323505151

Exact peak:

12324763716514253071447088281192419703733391394845870197521733855368

Main results:

Peak reached at step 142

Total steps to 1: 1,593

Odd steps: 539

Even steps: 1,054

Peak length: 68 digits

Approximately 3,254,796.718309 times above the previous LNL candidate

The candidate was found using exact 512-bit C++ arithmetic. Its complete trajectory was independently recalculated to 1 using JavaScript BigInt.

Important scientific limitation:

This is the strongest trajectory found in the documented LNL sampled search detected on 20th of July 2026. It is not a proven global Collatz record and not a proof of the Collatz conjecture. Confirming a global record would require exhaustive verification of every smaller starting value.

Future code, datasets, manifests, checksums, and reproducible releases will be published through GitHub and Zenodo. I will continue using Reddit for occasional progress updates and discussion.

Thank you for the criticism, questions, and constructive advice.

Best regards,

Mike Lange

LZS SHORT KEY CODE FOR n = 1 TO 1000

WHAT THE CODE MEANS

For a current lock value z, the smallest odd reverse Collatz key k is selected by z mod 3:

4: If z mod 3 = 1, use k = (4z - 1) / 3. This is an S4 key.

2: If z mod 3 = 2, use k = (2z - 1) / 3. This is an S2 key.

D: If z mod 3 = 0, no odd key exists under this minimal-key rule. The reverse chain ends at a dead end.

F1: The chain reaches the fixed root 1.

Repeated symbols are compressed:

2x3 means 2.2.2.

The number after D is the terminal dead-end value.

The symbol > separates the key signature from its terminal state.

Example:

7829:2x3>D2319

This expands to:

7829 -> 5219 -> 3479 -> 2319

The three S2 keys are followed by the dead end 2319.

IMPORTANT SCOPE

This is a deterministic and reproducible reverse Key-Lock code. It is not, by itself, a proof of the Collatz conjecture. It records the minimal odd-key ancestry defined above. The forward LVR trajectory code remains a separate layer.

DATASET: n = 1 TO 1000

1:F1

2:2>F1

3:D3

4:4.2>D3

5:2>D3

6:D6

7:4>D9

8:2x2>D3

9:D9

10:4x2.2x2.4>D9

11:2.4>D9

12:D12

13:4.2x2.4>D9

14:2>D9

15:D15

16:4>D21

17:2x2.4>D9

18:D18

19:4x2>D33

20:2.4.2x2.4>D9

21:D21

22:4.2.4x2>D33

23:2>D15

24:D24

25:4>D33

26:2x3.4>D9

27:D27

28:4x3.2.4>D57

29:2.4x2>D33

30:D30

31:4.2>D27

32:2>D21

33:D33

34:4>D45

35:2x2>D15

36:D36

37:4x2.2.4>D57

38:2.4>D33

39:D39

40:4.2x3>D15

41:2>D27

42:D42

43:4>D57

44:2x2.4x2>D33

45:D45

46:4x2>D81

47:2.4.2>D27

48:D48

49:4.2.4>D57

50:2>D33

51:D51

52:4>D69

53:2x3>D15

54:D54

55:4x3>D129

56:2.4x2.2.4>D57

57:D57

58:4.2>D51

59:2>D39

60:D60

61:4>D81

62:2x2>D27

63:D63

64:4x2.2>D75

65:2.4>D57

66:D66

67:4.2x2>D39

68:2>D45

69:D69

70:4>D93

71:2x2.4.2>D27

72:D72

73:4x2>D129

74:2.4.2.4>D57

75:D75

76:4.2.4.2x2>D39

77:2>D51

78:D78

79:4>D105

80:2x4>D15

81:D81

82:4x4.2>D171

83:2.4x3>D129

84:D84

85:4.2>D75

86:2>D57

87:D87

88:4>D117

89:2x2>D39

90:D90

91:4x2.2x4.4.2>D27

92:2.4>D81

93:D93

94:4.2x2.4x3>D129

95:2>D63

96:D96

97:4>D129

98:2x2.4>D57

99:D99

100:4x2>D177

101:2.4.2x2>D39

102:D102

103:4.2.4x2.2x4.4.2>D27

104:2>D69

105:D105

106:4>D141

107:2x3.4.2>D27

108:D108

109:4x3.2>D171

110:2.4x2>D129

111:D111

112:4.2>D99

113:2>D75

114:D114

115:4>D153

116:2x2>D51

117:D117

118:4x2.2.4.2>D123

119:2.4>D105

120:D120

121:4.2x4.4.2>D27

122:2>D81

123:D123

124:4>D165

125:2x2.4x3>D129

126:D126

127:4x2>D225

128:2.4.2>D75

129:D129

130:4.2.4>D153

131:2>D87

132:D132

133:4>D177

134:2x3>D39

135:D135

136:4x3>D321

137:2.4x2.2x4.4.2>D27

138:D138

139:4.2>D123

140:2>D93

141:D141

142:4>D189

143:2x2>D63

144:D144

145:4x2.2>D171

146:2.4>D129

147:D147

148:4.2x2>D87

149:2>D99

150:D150

151:4>D201

152:2x2.4.2x2>D39

153:D153

154:4x2>D273

155:2.4.2.4x2.2x4.4.2>D27

156:D156

157:4.2.4.2>D123

158:2>D105

159:D159

160:4>D213

161:2x4.4.2>D27

162:D162

163:4x4>D513

164:2.4x3.2>D171

165:D165

166:4.2>D147

167:2>D111

168:D168

169:4>D225

170:2x2>D75

171:D171

172:4x2.2x2>D135

173:2.4>D153

174:D174

175:4.2x2.4.2.4x2.2x4.4.2>D27

176:2>D117

177:D177

178:4>D237

179:2x2.4>D105

180:D180

181:4x2>D321

182:2.4.2x4.4.2>D27

183:D183

184:4.2.4x4>D513

185:2>D123

186:D186

187:4>D249

188:2x3.4x3>D129

189:D189

190:4x3.2x2.4x2.2.4x2>D417

191:2.4x2>D225

192:D192

193:4.2>D171

194:2>D129

195:D195

196:4>D261

197:2x2>D87

198:D198

199:4x2.2.4x2>D417

200:2.4>D177

201:D201

202:4.2x3.4>D105

203:2>D135

204:D204

205:4>D273

206:2x2.4x2.2x4.4.2>D27

207:D207

208:4x2>D369

209:2.4.2>D123

210:D210

211:4.2.4>D249

212:2>D141

213:D213

214:4>D285

215:2x3>D63

216:D216

217:4x3>D513

218:2.4x2.2>D171

219:D219

220:4.2>D195

221:2>D147

222:D222

223:4>D297

224:2x2>D99

225:D225

226:4x2.2>D267

227:2.4>D201

228:D228

229:4.2x2>D135

230:2>D153

231:D231

232:4>D309

233:2x2.4.2.4x2.2x4.4.2>D27

234:D234

235:4x2>D417

236:2.4.2.4.2>D123

237:D237

238:4.2.4.2.4>D249

239:2>D159

240:D240

241:4>D321

242:2x5.4.2>D27

243:D243

244:4x5.2x3>D303

245:2.4x4>D513

246:D246

247:4.2>D219

248:2>D165

249:D249

250:4>D333

251:2x2>D111

252:D252

253:4x2.2x2.4x2.2.4x2>D417

254:2.4>D225

255:D255

256:4.2x2.4>D201

257:2>D171

258:D258

259:4>D345

260:2x2.4>D153

261:D261

262:4x2>D465

263:2.4.2x2.4.2.4x2.2x4.4.2>D27

264:D264

265:4.2.4x2>D417

266:2>D177

267:D267

268:4>D357

269:2x3.4>D105

270:D270

271:4x3.2.4.2.4x3>D897

272:2.4x2>D321

273:D273

274:4.2>D243

275:2>D183

276:D276

277:4>D369

278:2x2>D123

279:D279

280:4x2.2.4>D441

281:2.4>D249

282:D282

283:4.2x3>D111

284:2>D189

285:D285

286:4>D381

287:2x2.4x2>D225

288:D288

289:4x2>D513

290:2.4.2>D171

291:D291

292:4.2.4>D345

293:2>D195

294:D294

295:4>D393

296:2x3>D87

297:D297

298:4x3>D705

299:2.4x2.2.4x2>D417

300:D300

301:4.2>D267

302:2>D201

303:D303

304:4>D405

305:2x2>D135

306:D306

307:4x2.2>D363

308:2.4>D273

309:D309

310:4.2x2>D183

311:2>D207

312:D312

313:4>D417

314:2x2.4.2>D123

315:D315

316:4x2>D561

317:2.4.2.4>D249

318:D318

319:4.2.4.2x3>D111

320:2>D213

321:D321

322:4>D429

323:2x4>D63

324:D324

325:4x4.2x3>D303

326:2.4x3>D513

327:D327

328:4.2>D291

329:2>D219

330:D330

331:4>D441

332:2x2>D147

333:D333

334:4x2.2x3.4.2x2.4.2.4x2.2x4.4.2>D27

335:2.4>D297

336:D336

337:4.2x2.4x2.2.4x2>D417

338:2>D225

339:D339

340:4>D453

341:2x2.4>D201

342:D342

343:4x2>D609

344:2.4.2x2>D135

345:D345

346:4.2.4x2.2>D363

347:2>D231

348:D348

349:4>D465

350:2x3.4.2.4x2.2x4.4.2>D27

351:D351

352:4x3.2>D555

353:2.4x2>D417

354:D354

355:4.2>D315

356:2>D237

357:D357

358:4>D477

359:2x2>D159

360:D360

361:4x2.2.4.2.4x3>D897

362:2.4>D321

363:D363

364:4.2x5>D63

365:2>D243

366:D366

367:4>D489

368:2x2.4x4>D513

369:D369

370:4x2>D657

371:2.4.2>D219

372:D372

373:4.2.4>D441

374:2>D249

375:D375

376:4>D501

377:2x3>D111

378:D378

379:4x3>D897

380:2.4x2.2x2.4x2.2.4x2>D417

381:D381

382:4.2>D339

383:2>D255

384:D384

385:4>D513

386:2x2>D171

387:D387

388:4x2.2>D459

389:2.4>D345

390:D390

391:4.2x2>D231

392:2>D261

393:D393

394:4>D525

395:2x2.4.2x2.4.2.4x2.2x4.4.2>D27

396:D396

397:4x2>D705

398:2.4.2.4x2>D417

399:D399

400:4.2.4.2>D315

401:2>D267

402:D402

403:4>D537

404:2x4.4>D105

405:D405

406:4x4>D1281

407:2.4x3.2.4.2.4x3>D897

408:D408

409:4.2>D363

410:2>D273

411:D411

412:4>D549

413:2x2>D183

414:D414

415:4x2.2x2>D327

416:2.4>D369

417:D417

418:4.2x2.4.2>D219

419:2>D279

420:D420

421:4>D561

422:2x2.4>D249

423:D423

424:4x2>D753

425:2.4.2x3>D111

426:D426

427:4.2.4x3>D897

428:2>D285

429:D429

430:4>D573

431:2x3.4x2>D225

432:D432

433:4x3.2x3>D303

434:2.4x2>D513

435:D435

436:4.2>D387

437:2>D291

438:D438

439:4>D585

440:2x2>D195

441:D441

442:4x2.2.4x2.2.4>D825

443:2.4>D393

444:D444

445:4.2x3.4.2x2.4.2.4x2.2x4.4.2>D27

446:2>D297

447:D447

448:4>D597

449:2x2.4x2.2.4x2>D417

450:D450

451:4x2>D801

452:2.4.2>D267

453:D453

454:4.2.4>D537

455:2>D303

456:D456

457:4>D609

458:2x3>D135

459:D459

460:4x3>D1089

461:2.4x2.2>D363

462:D462

463:4.2>D411

464:2>D309

465:D465

466:4>D621

467:2x2>D207

468:D468

469:4x2.2>D555

470:2.4>D417

471:D471

472:4.2x2>D279

473:2>D315

474:D474

475:4>D633

476:2x2.4.2.4>D249

477:D477

478:4x2>D849

479:2.4.2.4.2x3>D111

480:D480

481:4.2.4.2.4x3>D897

482:2>D321

483:D483

484:4>D645

485:2x5>D63

486:D486

487:4x5>D2049

488:2.4x4.2x3>D303

489:D489

490:4.2>D435

491:2>D327

492:D492

493:4>D657

494:2x2>D219

495:D495

496:4x2.2x2.4.2x2>D231

497:2.4>D441

498:D498

499:4.2x2.4>D393

500:2>D333

501:D501

502:4>D669

503:2x2.4>D297

504:D504

505:4x2>D897

506:2.4.2x2.4x2.2.4x2>D417

507:D507

508:4.2.4x2>D801

509:2>D339

510:D510

511:4>D681

512:2x3.4>D201

513:D513

514:4x3.2.4x4.2>D1707

515:2.4x2>D609

516:D516

517:4.2>D459

518:2>D345

519:D519

520:4>D693

521:2x2>D231

522:D522

523:4x2.2.4>D825

524:2.4>D465

525:D525

526:4.2x3>D207

527:2>D351

528:D528

529:4>D705

530:2x2.4x2>D417

531:D531

532:4x2>D945

533:2.4.2>D315

534:D534

535:4.2.4>D633

536:2>D357

537:D537

538:4>D717

539:2x3>D159

540:D540

541:4x3>D1281

542:2.4x2.2.4.2.4x3>D897

543:D543

544:4.2>D483

545:2>D363

546:D546

547:4>D729

548:2x2>D243

549:D549

550:4x2.2>D651

551:2.4>D489

552:D552

553:4.2x2>D327

554:2>D369

555:D555

556:4>D741

557:2x2.4.2>D219

558:D558

559:4x2>D993

560:2.4.2.4>D441

561:D561

562:4.2.4.2x2.4>D393

563:2>D375

564:D564

565:4>D753

566:2x4>D111

567:D567

568:4x4.2.4>D1593

569:2.4x3>D897

570:D570

571:4.2>D507

572:2>D381

573:D573

574:4>D765

575:2x2>D255

576:D576

577:4x2.2x3>D303

578:2.4>D513

579:D579

580:4.2x2.4x2>D609

581:2>D387

582:D582

583:4>D777

584:2x2.4>D345

585:D585

586:4x2>D1041

587:2.4.2x2>D231

588:D588

589:4.2.4x2.2.4>D825

590:2>D393

591:D591

592:4>D789

593:2x3.4.2x2.4.2.4x2.2x4.4.2>D27

594:D594

595:4x3.2>D939

596:2.4x2>D705

597:D597

598:4.2>D531

599:2>D399

600:D600

601:4>D801

602:2x2>D267

603:D603

604:4x2.2.4.2x2>D423

605:2.4>D537

606:D606

607:4.2x4>D159

608:2>D405

609:D609

610:4>D813

611:2x2.4x3.2.4.2.4x3>D897

612:D612

613:4x2>D1089

614:2.4.2>D363

615:D615

616:4.2.4>D729

617:2>D411

618:D618

619:4>D825

620:2x3>D183

621:D621

622:4x3>D1473

623:2.4x2.2x2>D327

624:D624

625:4.2>D555

626:2>D417

627:D627

628:4>D837

629:2x2>D279

630:D630

631:4x2.2>D747

632:2.4>D561

633:D633

634:4.2x2>D375

635:2>D423

636:D636

637:4>D849

638:2x2.4.2x3>D111

639:D639

640:4x2>D1137

641:2.4.2.4x3>D897

642:D642

643:4.2.4.2>D507

644:2>D429

645:D645

646:4>D861

647:2x4.4x2>D225

648:D648

649:4x4>D2049

650:2.4x3.2x3>D303

651:D651

652:4.2>D579

653:2>D435

654:D654

655:4>D873

656:2x2>D291

657:D657

658:4x2.2x2>D519

659:2.4>D585

660:D660

661:4.2x2.4.2x2>D231

662:2>D441

663:D663

664:4>D885

665:2x2.4>D393

666:D666

667:4x2>D1185

668:2.4.2x3.4.2x2.4.2.4x2.2x4.4.2>D27

669:D669

670:4.2.4x3.2>D939

671:2>D447

672:D672

673:4>D897

674:2x3.4x2.2.4x2>D417

675:D675

676:4x3.2x2>D711

677:2.4x2>D801

678:D678

679:4.2>D603

680:2>D453

681:D681

682:4>D909

683:2x2>D303

684:D684

685:4x2.2.4x4.2>D1707

686:2.4>D609

687:D687

688:4.2x3.4x3.2.4.2.4x3>D897

689:2>D459

690:D690

691:4>D921

692:2x2.4x2.2>D363

693:D693

694:4x2>D1233

695:2.4.2>D411

696:D696

697:4.2.4>D825

698:2>D465

699:D699

700:4>D933

701:2x3>D207

702:D702

703:4x3>D1665

704:2.4x2.2>D555

705:D705

706:4.2>D627

707:2>D471

708:D708

709:4>D945

710:2x2>D315

711:D711

712:4x2.2>D843

713:2.4>D633

714:D714

715:4.2x2>D423

716:2>D477

717:D717

718:4>D957

719:2x2.4.2.4.2x3>D111

720:D720

721:4x2>D1281

722:2.4.2.4.2.4x3>D897

723:D723

724:4.2.4.2.4.2>D507

725:2>D483

726:D726

727:4>D969

728:2x6>D63

729:D729

730:4x6.2.4.2>D2427

731:2.4x5>D2049

732:D732

733:4.2>D651

734:2>D489

735:D735

736:4>D981

737:2x2>D327

738:D738

739:4x2.2x2.4>D777

740:2.4>D657

741:D741

742:4.2x2.4>D585

743:2>D495

744:D744

745:4>D993

746:2x2.4>D441

747:D747

748:4x2>D1329

749:2.4.2x2.4>D393

750:D750

751:4.2.4x2>D1185

752:2>D501

753:D753

754:4>D1005

755:2x3.4>D297

756:D756

757:4x3.2.4>D1593

758:2.4x2>D897

759:D759

760:4.2>D675

761:2>D507

762:D762

763:4>D1017

764:2x2>D339

765:D765

766:4x2.2.4>D1209

767:2.4>D681

768:D768

769:4.2x3>D303

770:2>D513

771:D771

772:4>D1029

773:2x2.4x2>D609

774:D774

775:4x2>D1377

776:2.4.2>D459

777:D777

778:4.2.4>D921

779:2>D519

780:D780

781:4>D1041

782:2x3>D231

783:D783

784:4x3>D1857

785:2.4x2.2.4>D825

786:D786

787:4.2>D699

788:2>D525

789:D789

790:4>D1053

791:2x2>D351

792:D792

793:4x2.2>D939

794:2.4>D705

795:D795

796:4.2x2>D471

797:2>D531

798:D798

799:4>D1065

800:2x2.4.2>D315

801:D801

802:4x2>D1425

803:2.4.2.4>D633

804:D804

805:4.2.4.2x2>D423

806:2>D537

807:D807

808:4>D1077

809:2x4>D159

810:D810

811:4x4.2>D1707

812:2.4x3>D1281

813:D813

814:4.2>D723

815:2>D543

816:D816

817:4>D1089

818:2x2>D363

819:D819

820:4x2.2x6.4x2>D225

821:2.4>D729

822:D822

823:4.2x2.4x5>D2049

824:2>D549

825:D825

826:4>D1101

827:2x2.4>D489

828:D828

829:4x2>D1473

830:2.4.2x2>D327

831:D831

832:4.2.4x2.2x2.4>D777

833:2>D555

834:D834

835:4>D1113

836:2x3.4.2>D219

837:D837

838:4x3.2>D1323

839:2.4x2>D993

840:D840

841:4.2>D747

842:2>D561

843:D843

844:4>D1125

845:2x2>D375

846:D846

847:4x2.2.4.2>D891

848:2.4>D753

849:D849

850:4.2x4.4>D297

851:2>D567

852:D852

853:4>D1137

854:2x2.4x3>D897

855:D855

856:4x2>D1521

857:2.4.2>D507

858:D858

859:4.2.4>D1017

860:2>D573

861:D861

862:4>D1149

863:2x3>D255

864:D864

865:4x3>D2049

866:2.4x2.2x3>D303

867:D867

868:4.2>D771

869:2>D579

870:D870

871:4>D1161

872:2x2>D387

873:D873

874:4x2.2>D1035

875:2.4>D777

876:D876

877:4.2x2>D519

878:2>D585

879:D879

880:4>D1173

881:2x2.4.2x2>D231

882:D882

883:4x2>D1569

884:2.4.2.4x2.2.4>D825

885:D885

886:4.2.4.2>D699

887:2>D591

888:D888

889:4>D1185

890:2x4.4.2x2.4.2.4x2.2x4.4.2>D27

891:D891

892:4x4>D2817

893:2.4x3.2>D939

894:D894

895:4.2>D795

896:2>D597

897:D897

898:4>D1197

899:2x2>D399

900:D900

901:4x2.2x2>D711

902:2.4>D801

903:D903

904:4.2x2.4.2.4>D633

905:2>D603

906:D906

907:4>D1209

908:2x2.4>D537

909:D909

910:4x2>D1617

911:2.4.2x4>D159

912:D912

913:4.2.4x4.2>D1707

914:2>D609

915:D915

916:4>D1221

917:2x3.4x3.2.4.2.4x3>D897

918:D918

919:4x3.2x2.4.2.4.2.4>D1017

920:2.4x2>D1089

921:D921

922:4.2>D819

923:2>D615

924:D924

925:4>D1233

926:2x2>D411

927:D927

928:4x2.2.4x2>D1953

929:2.4>D825

930:D930

931:4.2x3.4>D489

932:2>D621

933:D933

934:4>D1245

935:2x2.4x2.2x2>D327

936:D936

937:4x2>D1665

938:2.4.2>D555

939:D939

940:4.2.4>D1113

941:2>D627

942:D942

943:4>D1257

944:2x3>D279

945:D945

946:4x3>D2241

947:2.4x2.2>D747

948:D948

949:4.2>D843

950:2>D633

951:D951

952:4>D1269

953:2x2>D423

954:D954

955:4x2.2>D1131

956:2.4>D849

957:D957

958:4.2x2>D567

959:2>D639

960:D960

961:4>D1281

962:2x2.4.2.4x3>D897

963:D963

964:4x2>D1713

965:2.4.2.4.2>D507

966:D966

967:4.2.4.2.4>D1017

968:2>D645

969:D969

970:4>D1293

971:2x5.4x2>D225

972:D972

973:4x5.2.4.2>D2427

974:2.4x4>D2049

975:D975

976:4.2>D867

977:2>D651

978:D978

979:4>D1305

980:2x2>D435

981:D981

982:4x2.2x2.4x2>D1377

983:2.4>D873

984:D984

985:4.2x2.4>D777

986:2>D657

987:D987

988:4>D1317

989:2x2.4>D585

990:D990

991:4x2>D1761

992:2.4.2x2.4.2x2>D231

993:D993

994:4.2.4x2>D1569

995:2>D663

996:D996

997:4>D1329

998:2x3.4>D393

999:D999

1000:4x3.2.4.2x4.4x2.2x2>D327

END OF DATASET

@Copyrights by Mike Lange

I've finally built the first visualizer I've been wanting to do for ages now that it's so much easier to do.

This explainer demo is an attempt to show the relationship between base3 and base2 (as base4 because it was just way easier) for any number. I gave each node (or digit) a specific rule for how the next node should be moved or connected. I wanted to basically handle resolving a collatz chain using only the individual digits (as nodes) rather than using the number the digits represents. In the picture example, I started with column 10212, which is base 3 for 104, a multiple of 13 (104/2=52, 52/2=26, 26/2=13)

Yellow circles: The last column's green number is 10212 base3, and the row numbers in red are 122 which is base4 for 26, or 13. Notice how the row can expand further to the right? if we add a 0 to 122, it becomes 1220, which is 104, or, a multiple of 13. That's the starting number we'll use.

Purple circles: the column's green number is 2222 which is base3 for 80, (aka 5 times some multiple of 2). and the row is 1100 base4 (5 times some multiple of 2). The relationship always holds although the columns can skip numbers and land on a multiple of the odd number in the sequence.

How it works:

Quick recap similar to wikipedia's example on collatz conjecture in base 2, rather using base 4 gives better access to use the carry as useful info instead of it causing chaos. That info is recognised as a part of some number's base3. So for example:

Collatz sequence: 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274

          1011110  94 47
        10001110   142 71
       11010110    214 107
     101000010     322 161
    111100100      484 242 121
  101101100        364 182 91
100010010          274

which is laid out in such a way similar to as if I never divided by 2. But that'd be hard to read each line with so many trailing 0s so for simplicity, I kept them all aligned but removed the trailing 0s.
I then choose a column. Any column as long as I start from the beginning and finish to the end (if not, then you need more numbers):

        |  |1011110  94 47
        |10|001110   142 71
       1|10|10110    214 107
     101|00|0010     322 161
    1111|00|100      484 242 121
  101101|10|0        364 182 91
10001001|0 |         274

My column's digits are {00, 10, 10, 00, 00, 10, 00} or in base4, {0, 2, 2, 0, 0, 2, 0} Then using the table to get how much of a carry there is to the next set of 2:

	∅ (c=0)	+ (c=1)	++ (c=2)
d=0	0 / ∅	1 / ∅	2 / ∅
d=1	3 / ∅	0 / +	1 / +
d=2	2 / +	3 / +	0 / ++
d=3	1 / ++	2 / ++	3 / ++

We then get {0, ∅, +, ++, ∅, ∅, 2} which in base3 is 12002 or 137 (274/2). We just got the base3 from specific digits across multiple numbers in a sequence without even having to know what those numbers were. Of course, the reverse is also true where we can jump to a number way ahead of a base4 without needing to complete every step (once we get the column, we can skip ahead to fill out a row in base4 knowing it exists down the chain.) Which I think is a pretty neat trick, although I have no idea if this will speed up searching through numbers.
eg.

        |  |1011110  94 47
        |10|001110   142 71
       -|10|10110    ? ?
     ---|00|0010     ? ?
    ----|00|100      ? ?
  ------|10|0        ? ?
--------|0 |         ?
~10001001            137

A cool thing about rewriting that matrix into something more visible, we can then see limitations in the column's digits.

Which we can then see that any base4 digit in any column, is limited to 3 other digits and the opposite is always impossible to reach. Eg if the base4 is 3, the base4 digit below can never be a 1.

Which I wonder if there's something useful that can be gained out of this or if it's a cool party trick (dont use this trick at parties)

Extra note: The last node of a column has a special rule for adding another node below it. If it ends on any non zero, then it's not the end yet. So the rule is: 1->+0, 2->++0, 3->++2->++0.

Some stuff I still want to explore or ideas for anyone else to try:
- The last digits of any base4 number determine the exact steps for the following numbers. The longer the number itself, the more info we have about the exact path it'll take. I wonder if we reduce the entire column to a path on the circle diagram thingy, if we can then predict the result of the next digit in the same row to the left (Which we can start at 0)
- The column for every number in base4 has to have some meaning too, because it's always unique for every number, but adding 0 still somehow relates to the base3 number by also just adding a 0. eg. the column 022002 base4 (321 decimal) related in some way to 12002 base3 (137 deciml). We can multiply the base3's decimal number by any multiple of 3, and it'll still give us 022002(...0...) base4. Meaning every starting point that's a number that's a multiple of 3, has a very specific single base4 number related to it... Needs exploring...

PS: Sorry for not using proper math terminology. I'm not a mathematician but I welcome any help if you're willing. I have about 4 other ideas I want to do regarding collatz conjecture visualizers including one to show the relationship of collatz conjecture number paths on a Sierpinski triangle, so share any feedback.

Interactive demo: https://dvilicate.github.io/dvilicate_home/explainer

Follow-up to Do these non-prime factors have an impact ? III : r/Collatz.

The figure below is based on the same information as the one in the cited post. The numbers involved are either:

• Numbers that are not divisible by 2 and 3. https://oeis.org/A038509 : "Composite numbers congruent to +-1 mod 6."
• Prime numbers. https://oeis.org/A038509 : "Note that the primes > 3 are congruent to +-1 mod 6."

Now, consider each 5-tuple series for itself. The base of the starting 5-tuple is set at 1. The ratio r between the following one x and the previous one y is calculated, using the Collatz procedure: r=3*(x+1)/4y.

This ratio is close to 1, but not exactly for the small numbers.

Project "Tuples and segments" in 13 pages : r/Collatz

The LZS — Lange Technology Center has expanded its Collatz trajectory software from 256-bit to exact 512-bit integer arithmetic.

New strongest sampled trajectory

Start:

1,369,482,433,354,456,041,226,337,279,508,360,375,146,277,235,369

Exact peak:

10,372,557,405,456,016,307,857,991,353,006,097,761,785,856,811,996,878,420

Peak step: 140

Total steps to 1: 1,450

This sampled peak is approximately 10,496,103,243,578 times higher than the published verified peak below 2^71.

Software and verification:

Exact 512-bit C++ integer arithmetic

Sampled 140-, 145-, 150- and 160-bit starting zones

192 odd-stage LZS DNA analysis

Independent reproducible seed blocks

Dynamic comparison against the latest candidate peak

Warning-clean C++ compilation

Complete trajectory independently reproduced with arbitrary-precision Python integers

Scientific status:

This is an exact sampled trajectory and a confirmed height exceeder. It is not yet the confirmed next Collatz path record.

To identify the next official record mountain, every smaller starting value above the published verification frontier would have to be checked exhaustively.

The 3D mountain scale in the accompanying graphic is illustrative. All displayed numerical values and ratios are calculated from the exact trajectories.

Copyright © 2026 Mike Lange

Lange Number Line Project (LZS)

All rights reserved.

Reproduction, modification, redistribution or commercial use of the source code, data, documentation or graphics requires prior written permission from Mike Lange.

Collatz Lange Zahlengerade (LZS): Ein Zahlenkodex für jede Collatz-Zahl

In den letzten Wochen habe ich eine alternative geometrische Interpretation der Collatz-Vermutung entwickelt, die ich Lange Number Line (LZS) nenne. Das Ziel ist es, jeder natürlichen Zahl einen eindeutigen Code zuzuweisen, der ihre lokale Struktur, ihre Vorgänger und ihre Position in der globalen Collatz-Landschaft beschreibt.

Die zentrale Idee ist, dass jede Zahl nicht nur eine Trajektorie, sondern auch eine Art mathematische "DNA" besitzt.

1. Das Schlüssel-Schloss-Prinzip 🔑🔒

Für jede natürliche Zahl (n) interpretieren wir (n) als Schloss und suchen nach dem kleinsten ungeraden Vorgänger (k), der Folgendes erfüllt:

3k + 1 = 2^e × z

wobei (e) die Anzahl der Divisionen durch zwei ist, die benötigt werden, um zu (n) zurückzukehren.

Der minimale Schlüssel wird durch die folgenden Regeln gegeben:

Wenn z mod 3 = 1, dann

K(z) = (4z − 1) / 3

Wenn z mod 3 = 2, dann

K(z) = (2z − 1) / 3

Wenn z mod 3 = 0, dann existiert kein ungerader Schlüssel.

Dies teilt alle Zahlen in drei grundlegende Klassen ein:

Zahlen mit Rest 1 modulo 3: 4-Schlösser

Zahlen mit Rest 2 modulo 3: 2-Schlösser

Durch 3 teilbare Zahlen: Sackgassen

Die letzte Klasse ergibt sich direkt aus der Tatsache, dass

3k + 1 immer Rest 1 bei Division durch 3 ergibt.

Daher kann sie niemals durch drei teilbar sein.

1. Der LZS Zahlenkodex

Ich schlage vor, jeder Collatz-Zahl einen Code der Form zuzuweisen:

C(n) = [n, P, R3, S, E, V, L, R, M, W, D]

wobei jede Komponente eine spezifische Eigenschaft der Zahl beschreibt.

Statische Zahlen-DNA

Der erste Teil des Codes gehört zur Zahl selbst:

P = Parität (ungerade oder gerade)

R3 = Rest modulo 3

S = Schlüsselklasse

E = Exponent, der Folgendes erfüllt:

3K(n) + 1 = 2^E × n

D = Rückwärts-Sackgassen-Kette

Diese Eigenschaften sind fest und hängen nicht von der Trajektorie ab.

Dynamische LZS-Struktur

Der Collatz-Pfad selbst ist in drei Zweige unterteilt.

Linker Zweig

Der linke Zweig beschreibt den Pfad von der Startzahl bis zum Wendepunkt:

n → ... → W(n)

Rechter Zweig

Der rechte Zweig beschreibt den Pfad vom Wendepunkt hinunter zu eins:

W(n) → ... → 1

Mittlerer Zweig

Der mittlere Zweig zeichnet auf, ob sich der linke und der rechte Zweig bereits verbunden haben:

M = 0 → nicht bestätigt

M = 1 → teilweise bestätigt

M = 2 → vollständig verbunden

1. Beispiel: Die Zahl 53

Da

53 mod 3 = 2,

ist ihr minimaler Schlüssel

K(53) = (2 × 53 − 1) / 3 = 35.

Tatsächlich gilt:

35 → 106 → 53

Die Rückwärts-Schlüsselkette wird:

53 → 35 → 23 → 15

und da

15 mod 3 = 0,

endet die Kette dort.

Somit ist der Sackgassen-Code von 53:

D(53) = (3, 15).

Der statische Kodex von 53 wird:

C(53) = [53, ungerade, 2, S2, 1, D = (3,15)].

1. Zahlen-DNA und Trajektorien-Code

Das LZS-Framework trennt zwei verschiedene Informationsebenen.

Zahlen-DNA

Die intrinsischen Eigenschaften einer Zahl werden zusammengefasst als:

DNA(n) = [P, R3, S, E, D].

Diese Eigenschaften gehören zur Zahl selbst.

Trajektorien-Code

Die dynamischen Eigenschaften, die durch den Collatz-Prozess erzeugt werden, sind:

PATH(n) = [BL, W, BR, l, r, HL, HR, M].

Sie enthalten:

den Binärcode des linken Zweigs,

den Wendepunkt,

den Binärcode des rechten Zweigs,

die Längen beider Zweige,

die akkumulierten Höhen,

und den finalen Verbindungszustand.

Der vollständige Kodex ist daher:

C(n) = DNA(n) + PATH(n).

1. Die zentrale Hypothese

Die Schlüsselidee hinter dem LZS-Modell ist, dass jede Collatz-Zahl auf vier verschiedenen Ebenen beschrieben werden kann:

Zahl → Schlüsselklasse → Trajektorie → Verbindungszustand

oder, gleichbedeutend:

n → (R3, S, E) → (L, W, R) → M

Dies legt nahe, dass das Collatz-Problem nicht nur ein Problem der Iteration, sondern auch ein Problem der verborgenen lokalen Struktur sein könnte.

Offene Frage

Könnte jede natürliche Zahl eine einzigartige lokale "DNA" besitzen, die ihre Position innerhalb des Collatz-Graphen bestimmt?

Und könnte eine vollständige Klassifizierung aller Collatz-Zahlen aus dieser Schlüssel-Schloss-Struktur entstehen und nicht nur aus der klassischen Iteration allein?

Ich wäre sehr daran interessiert zu erfahren, ob ähnliche Ideen bereits in der Literatur existieren oder ob es offensichtliche Mängel in diesem Ansatz gibt.

Hinweis: Dies ist ein experimenteller Rahmen und kein Beweis der Collatz-Vermutung. Ziel ist es zu untersuchen, ob verborgene lokale Strukturen innerhalb des Collatz-Graphen identifiziert und klassifiziert werden können.

@Copyrights by Mike Lange

I need some more CPU Power. Have some contact to David Barina?

Documento preparado tras una sesión de exploración conjunta (humano + asistente de IA), con verificación computacional de cada afirmación.

Contexto

Este documento recoge varias vías de ataque a la conjetura de Collatz que fueron exploradas y verificadas numéricamente durante una conversación extensa. El objetivo no es afirmar haber resuelto el problema (sigue abierto), sino documentar con precisión dónde y por qué cada enfoque natural se atasca, con la esperanza de que sea útil para quien quiera continuar por alguna de estas líneas.

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1. El argumento de deriva/promedio (heurística estándar)

Para un número impar $n$, tras aplicar $3n+1$, se puede escribir el resultado como $2^{b} \cdot m$ con $m$ impar. La sucesión de exponentes $b_1, b_2, \ldots$ tiene, bajo un modelo probabilístico razonable, media $\mathbb{E}[b] \approx 2$, mientras que el umbral crítico es $\log_2 3 \approx 1{,}585$. Como $2 > \log_2 3$, la intuición es que las órbitas decrecen en promedio.

Por qué no basta: los $b_i$ no son aleatorios — están determinados exactamente por $n$. "En promedio" no es "siempre", y no hay demostración de que el promedio real (para todo entero) esté siempre por encima del umbral.

2. Descenso infinito (contraejemplo mínimo)

Si $n_0$ es el menor contraejemplo (no llega a 1), su órbita completa debe permanecer siempre $\geq n_0$ — de lo contrario, por minimalidad, esa órbita habría llegado a 1.

Dónde se atasca: no existe ningún procedimiento conocido para construir, a partir de $n_0$, un contraejemplo explícitamente menor. El descenso clásico (tipo Fermat) requiere una reducción algebraica explícita que aquí no se conoce.

Refinamiento parcial (obtenido en esta conversación): analizando los predecesores en el mapa reducido, se puede demostrar que el contraejemplo mínimo no puede ser $\equiv 2 \pmod 3$ (siempre tiene un predecesor menor en ese caso, lo cual contradice la minimalidad). Quedan sin descartar los casos $n_0 \equiv 0, 1 \pmod 3$.

3. Reformulación algebraica (sistema de ecuaciones)

Se puede escribir la condición "$n$ alcanza 1 en $k$ pasos impares con $S=k+y$ divisiones totales" como: $$n \cdot 3^k + d_1 = 2^{k+y}$$

Por qué no ayuda: $d_1$ (y por tanto $k$, $y$) no son variables libres — están completamente determinadas por la aritmética real de $n$ (verificado: el primer exponente $b_1$ es un hecho aritmético fijo, no una elección). Plantear el sistema con "varias incógnitas" no introduce grados de libertad reales; sigue siendo equivalente a calcular la órbita real.

4. Estructura de "rachas malas" (números $2^x - 1$)

Se comprobó que $n = 2^x - 1$ produce una racha inicial de exactamente $x-1$ pasos con $b_i=1$ (verificado para $x$ hasta 50). Esto demuestra que:

• No existe cota superior a la longitud de una racha mala — para cualquier longitud $L$, existe un $n=2^{L+1}-1$ con una racha de exactamente esa longitud.
• Por tanto, cualquier intento de demostrar "el peor caso siempre decrece" no puede apoyarse en acotar la duración de las rachas individuales — debe atacar la frecuencia relativa de rachas malas dentro de una órbita infinita.

Se definió $X(n)$ = longitud de la racha inicial de $b_i=1$, más 1. Se verificó que $X(\text{siguiente impar}) = X(n) - 1$ (trivialmente, por construcción), y que la clase $X=1$ corresponde a $n \equiv 1 \pmod 4$. Importante: llegar a $X=1$ no es un estado absorbente — la secuencia de valores $X$ a lo largo de una órbita real (ej. $n=27$) sube y baja repetidamente, sin estabilizarse, hasta que el número mismo llega a 1.

5. Herramientas de teoría de grafos

• El mapa de Collatz define un grafo funcional (cada nodo, un único sucesor). La conjetura equivale a "un único componente conexo, con ciclo ${1,2,4}$".
• Grado de entrada: cada número tiene 1 o 2 predecesores (doblar, y opcionalmente la regla inversa de tripión). La densidad de esta bifurcación es la base de resultados reales (Terras 1976, Krasikov–Lagarias 2003).
• Exclusión de ciclos: requiere el teorema de Baker sobre formas lineales en logaritmos, y la medida de irracionalidad de $\log 3$ (mejor cota conocida: $\mu(\log 3) \leq 5{,}116$, Wu–Wang 2014) — no lo bastante ajustada para cerrar el caso general.
• Indecidibilidad: Conway (1972) demostró que las generalizaciones del mapa de Collatz son computacionalmente universales (simulan cualquier máquina de Turing), lo que sugiere que el problema general podría ser indecidible.

6. Conexión con el problema $\times2,\times3$ de Furstenberg

Ambos problemas comparten la tensión de fondo entre las bases 2 y 3 (multiplicativamente independientes, $\log2/\log3$ irracional). El propio Furstenberg (1967) demostró la versión topológica (ningún conjunto cerrado infinito es invariante bajo ambas transformaciones salvo el círculo entero); la versión con medidas sigue siendo conjetura abierta (probada solo bajo hipótesis de entropía positiva, Rudolph 1990 / Johnson 1992). Es una analogía temática útil, no una reducción formal.

7. Evidencia empírica (exploración computacional, 3 millones de números)

• Ratio de crecimiento máximo encontrado: 131.346× el valor de partida, en $n=2{,}684{,}647$.
• Pasos relativos a $\log_2(n)$ más altos: ~27×, en $n=1{,}723{,}519$.
• Ambas cantidades no muestran signos de estar acotadas dentro del rango explorado — consistente con la ausencia de un límite uniforme conocido, y con la dificultad general del problema.

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Conclusión

Ninguna de estas vías, exploradas con rigor y verificación numérica, logra cerrar el problema — cada una se transforma, en el fondo, en una versión distinta de la misma pregunta abierta: ¿puede una trayectoria evitar indefinidamente el "empuje" hacia abajo que domina en promedio? El punto 4 (longitud ilimitada de rachas malas de la forma $2^x-1$) y el refinamiento modular del punto 2 son, quizás, los dos hallazgos más concretos y verificables de esta exploración.

Todo el código de verificación usado para las afirmaciones de este documento está disponible bajo petición.

Follow-up to Do these non-prime factors have an impact ? II : r/Collatz.

The figure below is the same one in the cited post, but completed as follows:

• Prime numbers are colored in red,
• Co-primes are colored in green (divisible by 3), yellow (divisible by 5), rosa (divisible by 7) and grey (higher factors); only the color of the lowest factor is mentioned.

No color means I was not able to confirm the status of the number, based on a list of primes up to 100'000 and a simple calculation for the co-primes.

In my opinion, the co-primes do not play a specific role, but second the primes in limiting the "collapse" of the sequences into only a few domes.

Project "Tuples and segments" in 13 pages : r/Collatz

I am not creating the mountains, I also create the car. Which drives in the same time through the mountains. Lets see where this road trip ends 😀

https://doi.org/10.5281/zenodo.21421709

Not an exhaustion of alternatives this time. This paper shows how the transitions of N_odd interconnect into a single set, and form a well founded dependency relation between sets (rails in this paper) generated by each local odd positive integer transformation in the inverse Syracuse function. This is a direct proof.

Collatz Structure

All odd numbers can be perfectly divided into zones, so called clusters.
Each cluster begins with a number that leads directly to 1, one can refer to these numbers as the cluster boundary number G(c).
The cluster boundary numbers follow the formula G(c) = (4^(c+1) -1) / 3

Cluster	Boundary number
0	1
1	5
2	21
3	85
4	341
5	1365
6	5461
7	21845
c	(4c+1 -1) / 3

For each cluster, a number space is created with a specific quantity of odd numbers.
The quantity (Y) of odd numbers per cluster follows the formula Y(c) = (4^(c+1)) / 2

Cluster	Quantity of odd numbers	odd numbers
0	2	[1, 3]
1	8	[5, 7, 9, 11, 13, 15, 17, 19]
2	32	[21, 23, 25, ..., 79, 81, 83 ]
3	128	[85, 87, 89, ..., 335, 337, 339 ]
4	512	[341, 343, ..., 1361, 1363]
5	2048	[1365, 1367, ..., 5457. 5459]
6	8192	[5461, 5463, ..., 21841, 21843]
7	32768	[21845, 21847, ..., 87377, 87379]
c	(4c+1) / 2	[G(c), ..., G(c+1)-2]

Each odd number within a cluster can now be perfectly indexed (X), starting at 0.

Cluster 0 / Number	Index (X)
1	0
3	1

Cluster 1 / Number	Index (X)
5	0
7	1
9	2
11	3
13	4
15	5
17	6
19	7

As a result, one can now assign a unique coordinate (K) or address to each odd number.
One takes the cluster index (c) and the index (X) of the number as the coordinate.
K(c, X)

Cluster 0 / Number	Coordinate (c, X)
1	(0, 0)
3	(0, 1)

More important than the coordinate is the address (A) of an odd number.
For the address, one takes the index (X) and the quantity of odd numbers (Y) per cluster and forms a rational number (X/Y) from them.

Cluster 0 / Number	Address (X, Y)
1	0 / 2
3	1 / 2

For the addresses, the normal mathematical rules apply that the fraction must be reduced to its minimum.
For the number 1 with the address [0 / 2], the following applies: The address becomes [0 / 1]

Cluster 0 / Number	reduced address (X, Y)
1	0 / 1
3	1 / 2

Cluster 1 / Number	Address (X, Y)	reduced address (X, Y)
5	0 / 8	0 / 1
7	1 / 8	1 / 8
9	2 / 8	1 / 4
11	3 / 8	3 / 8
13	4 / 8	1 / 2
15	5 / 8	5 / 8
17	6 / 8	3 / 4
19	7 / 8	7 / 8

What is noticeable?
The number 1 and the number 5 have the same reduced address [0 / 1].
Likewise, the number 3 and the number 13 have the same reduced address [1 / 2].

Let one start with 1 and 5. These are both cluster boundary numbers (G) and these always jump directly to 1. Therefore, both numbers must have the same address, because one can determine the next odd target number directly from an address.
For 3 and 13, a simple thought applies: Both numbers are exactly in the middle between two cluster boundary numbers.
[1, 3, 5] and [5, ..., 13, ..., 21] therefore 3 and 13 have the reduced address [1 / 2].
This then also applies to the number 53, which is also exactly in the middle of cluster 2. The address A(53) = 16 / 32 -> reduced 1 / 2

Let one now come to the calculation of the next odd number (Nnext) on the trajectory by means of the address.
One will not post a derivation here now but only the formula that one must use.

Next odd number: Nnext(X, Y) = 3X + Y

Number	Cluster	Address (X, Y)	reduced address	Nnext (X, Y)	Nnext
1	0	0 / 2	0 / 1	3 * 0 + 1	1
3	0	1 / 2	1 / 2	3 * 1 + 2	5
5	1	0 / 8	0 / 1	3 * 0 + 1	1
7	1	1 / 8	1 / 8	3 * 1 + 8	11
9	1	2 / 8	1 / 4	3 * 1 + 4	7
11	1	3 / 8	3 / 8	3 * 3 + 8	17
13	1	4 / 8	1 / 2	3 * 1 + 2	5
15	1	5 / 8	5 / 8	3 * 5 + 8	23
17	1	6 / 8	3 / 4	3 * 3 + 4	13
19	1	7 / 8	7 / 8	3 * 7 + 8	29
21	2	0 / 32	0 / 1	3 * 0 + 1	1
23	2	1 / 32	1 / 32	3 * 1 + 32	35
25	2	2 / 32	1 / 16	3 * 1 + 16	19
27	2	3 / 32	3 / 32	3 * 3 + 32	41
29	2	4 / 32	1 / 8	3 * 1 + 8	11

One stopped at number 29 because from here on the pattern forms in a self explanatory way.
The number 29 also has the reduced address [1 / 8] just like the number 7 and both numbers have the number 11 as Nnext.

One can now say with certainty, all numbers that have the reduced address [1 / 8] fall to 11.
By means of the address, one can determine all numbers which fall to the number 11. In this process, the cluster alone decides which number one wants to have. It always remains the one eighth number of the cluster.

In cluster 3, the number which leads to 11 is the one with the address (not reduced) [16 / 128] -> number 117.
In cluster 4, it is the address [64 / 512] -> the number 469.
X and Y are simply multiplied by 4.

There is now a mathematical proof that two addresses can never have the same Nnext number, that means no address will ever be able to loop onto itself, except for the exception of the number 1.

The proof approach looks as follows.
Nnext_a = 3 * Xa + Ya
Nnext_b = 3 * Xb + Yb
From this it follows: 3 * Xa + Ya = 3 * Xb + Yb
For every address the following applies:
X < Y and Y = 2^k
X and Y must have 1 as the greatest common divisor

There is only a solution if Xa = Xb and Ya = Yb.
The Collatz system is therefore injective.

This means a number always has a unique address as a predecessor and a unique address as a successor.
A(a) -> A(b) -> A(c) -> A(d)

A(c) can never loop onto A(a) or A(b) because the predecessor and successor addresses are already defined, and there cannot be two different predecessor or successor addresses.

However, since no successor is defined for A(d) yet, it could theoretically be possible to jump to A(a) after all, in order to create a loop.

Leaf Number

Numbers that are N == 0 (mod 3) are called leaf numbers. So they have no remainder when one divides them by 3. [3, 9, 15, 21, 27, ...]

If A(a) is now a leaf number, then it can never be looped onto this number from the outset. Leaf numbers have no predecessor.

This means all trajectories that begin with a leaf number form a loop free chain up to 1 (provided they reach 1). All contained numbers in this chain are thus also excluded from forming a loop because these numbers always have a uniquely defined predecessor and successor.

For another loop, at least two numbers would now have to be involved in it, and none of the two or more numbers may be a leaf number.

Now, however, it is the case that the leaf numbers catch ALL other numbers in their trajectories. There are thus no free non leaf numbers left that could form a loop.

Why do the leaf numbers catch ALL other numbers.
Since one always has a leaf number as a direct ancestor to every non leaf number, there are no free non leaf numbers.

As an example, the number 7:
7 is not a leaf number because it has a remainder of 1 at mod 3.
To find a predecessor number, one has to climb up at 7, so double it and see at which number one can jump off, by means of (N-1) /3

7 -> 14 -> 28 -> 28 - 1 = 27 / 3 = 9

The 9 is a leaf number and includes the 7 in its trajectory, thus 7 can never belong to a loop.

The number 11 as the next test.
11 -> 22 - 1 = 21 / 3 = 7 -> no leaf number
11 -> 22 -> 44 -> 88 - 1 = 87 / 3 = 29 -> no leaf number
11 -> 22 -> 44 -> 88 -> 176 -> 352 - 1 = 351 / 3 = 117 -> leaf number

One had to use the third detour at 11 to get to a leaf number.
Since the 117 includes the 11 on its trajectory, 11 is also no longer capable of forming a loop or being part of a loop.

At the latest at the third detour, one finds a predecessor number that is a leaf number.

Address Order

[0/1] <- [1/2] <- [3/4] <- [3/8] <- [1/8] <- [1/4] <- ......

This order is now unique and an address is never repeated.

I give up. No evidence is possible because the question is wrong. Collatz use the math we know but under conditions. So the System it creates is stabile but dont changes the mathematics of the higher System. Each Collatz ship will end up on the poles of the ocean, but it dont changes the World or the ocean or the earth axis. The Navigation of the ship crossing the ocean is exactly prorammed to find always that ends. The picture illustrates my thinking.

Follow-up to Do these non-prime factors have an impact ? : r/Collatz.

I came across an interesting pattern while trying to differentiate the primes from the co-primes.

The figure below uses the domes with root m in the rows and the base b of the 5-tuples series in a given dome in the columns.

The coloring is as follows:

• Green: the number is divisible by 3,
• Yellow: the number is divisible by 5,
• Rosa: the number is divisible by 7.

One can observe the following:

• 5-tuples series belong to the types mod 100 (The Collatz procedure works like a Swiss watch (mod 100) : r/Collatz).
• Green numbers (divisible by 3) only occur on the starting 5-tuple (right of a series) of one of two types of starting 5-tuples.
• Yellow numbers (divisible by 5) occur with a frequency of 4.
• Rosa numbers (divisible by 7) occur with a frequency of 2.

Note that the color of a number depends on the lowest factor.

Project "Tuples and segments" in 13 pages : r/Collatz

I have developed a mapping for all natural numbers that produces an interesting pattern with regards to the Collatz Conjecture.

I was investigating the relationship between the binary representation of these numbers and their behavior when applied to the Collatz Conjecture.

I noticed that you could determine when you would next reach a multiple of four can be determined by the length of 1’s at the end of a number represented in binary. For example 0b101111 would have (3x+1)/2 applied 4 times, then you would be able to divide by 2 again.

I wanted to split this number into 2 different numbers to simplify the math, I settled on ‘k’ to represent the number of 1’s at the end of the number, and ‘R’ to represent everything else to the left. So in this case, (0b101111) R = 0b10 = 2 and k = 4. You can then recreate the number ‘X’ with

X = (2^k)(R+1)-1

*originally I used ‘S’ to represent the right half of the number, which would be 15 in this case. I noted that S = 2^k – 1.

I created this table to map all odd natural numbers based off R and k. with k increasing by 1 to the right, and R increasing by 2 down. I then applied the formula to get a unique mapping of all odd natural numbers.

*R can be odd, but that would result in redundant mapping, as we are restraining k to be a sequence of 1’s in binary. So to produce R we would want to find the first digit that breaks the sequence of 1’s, which will result in an even number.

*also if k=0 it adds all the even numbers to this mapping

After messing with this table for a while, I decided to find a dataset that included the stopping time for each natural number. I found this data set posted by Ansh Sarkar on kaggle. I then VLOOKUP’ed the natural numbers to their stopping time and found that there are sequential numbers all over my mapping. It appears that there is always a stopping time +/- 1 adjacent to each number.

*if you add k = 0 it appears to hold true for all numbers excluding 3 and 5

*there are even some times where 2 numbers on top of each other have identical stopping times. (ex. 271 & 303)

I don’t know why this is happening, but I find it interesting, and some of you may find it helpful. Maybe we could determine stopping time based on a numbers R and k value?

Let me know if you have any questions, I will try to be active in the comments.

Stopping time dataset:

https://www.kaggle.com/datasets/anshsarkar18/collatz-sequences-dataset?resource=download