Exponential diophantine problems

Gottfried Helms - Univ Kassel (Miniatures since 07' 2007)

 

Table for Fermat-quotients of degree m≥2
for integer bases b<p (where p
∈ ℙ )

The following table displays p-adic-expansions of bases b, for which

bp-1 ≡ 1 (mod pm)

In ([1],2009) I discussed earlier the problem of Fermat-quotients and introduced a couple of small tools and concepts which I use here. This table focuses on data for only a specific subset of the entire problem.

The casual reader who is not aware of the concept of Fermat-quotient may look in the wikipedia (see:[wp]) first which gives some basic insight and a couple of further references/online-links.

The reader may find some quite involved related tables in the internet, see the references-section at the end of this article for some online-links to homepages which I got aware of.

 

Some preliminary definitions:

1) In the following I (mis)use the braces for a shorter notation to express the above statement for a more general view. (A misuse which has been helpful through all my discussions/treatizes of exponential diophantine problems.) For the case, that a number n contains the primefactor p to the m'th power I write for p's canonical exponent in n:

{n, p} = m ==> n = x ∙ pm ∙with gcd(x,p)=1

Note that this notation is a bit more algebraic/concise than "pm| n" or "n≡0 (mod pm)"because -if in the two latter two cases m is the indeterminate- then the expression gives only an upper limit and not a precise value to m. So for instance,

{2p-1-1, p} = 2

is a bit more restricted/precise than the original definition for a Wieferich-prime p which requires only

{2p-1-1, p} 2

(where we know only two solutions p=1093 and p=3511 for m ≥2 anyway).

Similary for instance

{35-1 , 11} = 2
{19
6-1 , 7} = 3
{740862
6 1,7} = 7
{175909088838
12 1,13} = 12

indicate Fermat-quotients for (3, 11), (19, 7), (740862, 7) and (175909088838, 13) of degrees 2,3,7 and 12 respectively.

 

 

2) Let's define here the term "degree of the Fermat-quotient":

{bp-1-1, p } = m ==> m is the degree of the Fermat-quotient of the pair (b,p)

 

 

3) By the p-adic-representation of the bases b which provide high Fermat-quotients for some prime p we can read the table like in the following example:

p

b

d0

d1

d2

d3

d4

d4

d5

d6

d7

d8

d9

d10

...

7

1

1

0

0

0

0

0

0

0

0

0

0

0

...

7

2

2

4

6

3

0

2

6

2

4

3

4

4

...

7

3

3

4

6

3

0

2

6

2

4

3

4

4

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

 

Here the column p indicates the prime, the column b the "root" bases - "root" means here, that from b = d0 further bases bm can be derived according to the given p-adic digits d0,d1,d2,... Actually the entry in that column is identical with the entry in column d0 which simply contain all congruence-classes (mod p) .

The "p-adic" convention means, that the "digits" are just the digits of the number-representation of some number in the number-system to base p, so to say:

bm = "dm-1 ... d2 d1 d0" base p

so for the row with "root" base b=2 we can get the following subsequent bases b1,b2,...

b1 = 2 = 2 ==> {26 -1, 7} = 1
b2 = 2 + 4∙7 = 30 ==> {30
6-1, 7} = 2
b3 = 2 + 4∙7+ 6∙7
2 = 324 ==> {3246-1, 7} = 3
...
and in general
bm = 2 + 4∙7+ ... +dm-1∙7
m-1 ==> {bm6-1, 7} = m

But note, that also infinitely many bases of the form

let x a positive integer such that gcd(x,6) = 1
b2 = 2 + 4∙7+ x∙7
2 = 30+x∙49 ==> {b26-1, 7} = 2

give also fermat-quotients of degree 2, so

{796 - 1, 7} = 2

We can conclude one more result: because we see here a set of infinitely many bases in arithmetic progression, we have even - due to Dirichlet's theorem - a proof, that for each prime p and each degree m we'll find infinitely bases bm which are prime themselves.

So such a table essentially gives the (p-1)th roots of 1 modulo some pm in the p-adic sense and can in analogy be thought as representing the unit-roots exp(∙π∙m/(p-1)) in the case of the complex number plane.

(end of preliminary definitions)

 

Note that b=d0 being a residue (mod p) means, that all bases b gotten by p-adic-expansions beginning at a residue d0 <>0 (mod p) are also coprime to p and have no (yet recognizable) simple restrictions, so for instance they

* can be even,
* can be odd,
and in particular
* can be prime themselves.

Sometimes tabulations of Fermat-quotients display only prime bases b - for instance because there is great relevance connected to other exponential diophantine problems like FLT -for instance by the analyses of A. Wieferich (after whom that Wieferich-primes are named) and others. But this is not intended here; here I want to display the general scheme and in that scheme I want to focus to that subset of higher Fermat-quotients where the bases are in the residue classes of the associated primes and thus smaller that that primes.

 

Base b smaller than prime p

The example computation in the above lines shows immediately, that the displayed bases b = b1, simply equalling the residue-classes (mod p), are the only ones which are smaller than the prime which defines the congruence. High fermat-quotients are very rare for such cases; two prominent examples are the above mentioned two Wieferich primes p=1093 and p=3511 to base 2 where we have 210921 (mod 10932) and 235101 (mod 35112) or in the new notation {21092-1, 1093} = 2 and {23510-1, 3511} = 2 ; another one is p=11 and the base b=3 where we have {310 - 1, 11} = 2 .

To have such a case is equivalent to have the digit d1=0 in the table, because that means, according to the example above

b2 = d0 + 0∙p1 = d0 = b1 ==> {b1p-1 -1 , p} = 2

and the earliest example is that for prime p = 11 and base b1 = 3

 

p

b

d0

d1

d2

d3

d4

d4

d5

d6

d7

d8

d9

d10

d11

11

3

3

0

1

2

3

6

10

8

7

0

6

3

9

 

(Note that in general zeros in the digits occur randomly scattered over the digits in the table)

The well known Wieferich-primes p=1093 and p=3511 are immediately recognizable in the table by this: they are the only occurences so far, where the root base (= residue d0) equals 2 (or some small powers of 2) and the digit d1=0. They are marked orange here:

p

b

d0

d1

d2

d3

d4

d4

d5

d6

d7

d8

d9

d10

d11

1093

2

2

0

974

227

564

478

1009

395

581

1027

834

136

534

3511

2

2

0

102

2379

495

3468

1939

2849

2386

752

3077

803

3499

 

 

 

Higher degrees of Fermatquotiens

For even higher Fermat-quotients we can proceed by looking for further consecutive zeros in the digits:

b3 = d0 + 0∙p1+ 0∙p2 = b0 ==> {b3p-1 - 1, p} = 3
bm = d0 + 0∙p
1+ ...+ 0∙pm-1= b0 ==> {bmp-1 - 1, p} = m
...

 

Cases m=3

The only case of the fermat-quotient of order m=3 (where also the base is smaller than the defining prime) occurs with the prime p=113 and the base b3 = b1 = d0 = 68 :

p

b

d0

d1

d2

d3

d4

d4

d5

d6

d7

d8

d9

d10

d11

113

68

68

0

0

92

4

67

65

0

13

7

106

20

16

 

For this I checked the primes up to p=17 389 (which is the 2 000th prime). In the mathematical-discussionboard MSE (see[2]) one correspondent confirmed that observation and proceeded saying he had found this to be true up to p=104 729, (which is the 10 000'th prime) . In [3] we find that no further example occurs up to p ~ 3.6 ∙ 108

 

Cases m>3

I've not found any higher degree Fermat-quotients for pairs of bases and primes with b<p.

 

Gottfried Helms, 7.12.2013

 

Table 1, b<p:

p-adic representations for bases allowing fermat-quotients up to degree 12, where the high fermat-quotient occurs already for the "root" bases (which are smaller than the defining prime). That is equivalent to say that the p-adic-digit d1 = 0 .

p

b

d0

d1

d2

d3

d4

d4

d5

d6

d7

d8

d9

d10

d11

11

3

3

0

1

2

3

6

10

8

7

0

6

3

9

11

9

9

0

6

1

9

8

7

1

3

2

0

0

6

29

14

14

0

6

9

19

22

8

24

4

3

19

16

25

37

18

18

0

26

16

3

34

26

35

26

24

18

24

1

43

19

19

0

34

35

10

37

7

26

6

41

34

4

21

59

53

53

0

28

27

40

50

43

2

38

17

21

45

38

71

11

11

0

9

23

19

43

34

58

41

12

49

52

48

71

26

26

0

19

48

7

70

7

9

25

60

45

27

29

79

31

31

0

22

69

9

4

0

69

27

67

11

51

42

97

53

53

0

87

20

43

33

35

56

75

38

17

7

71

103

43

43

0

69

94

23

81

59

78

91

45

75

19

9

109

96

96

0

63

53

40

18

51

44

40

99

57

62

83

113

68

68

0

0

92

4

67

65

0

13

7

106

20

16

127

38

38

0

14

80

89

79

92

87

33

49

34

108

78

127

62

62

0

38

18

74

120

88

26

85

117

22

102

121

131

58

58

0

58

45

43

35

118

39

49

117

20

67

93

131

111

111

0

100

6

68

71

120

87

121

113

124

2

1

137

19

19

0

32

71

56

130

42

88

50

33

95

120

36

151

78

78

0

24

93

15

23

57

16

50

119

20

54

29

163

65

65

0

117

74

136

122

122

90

95

161

79

119

141

163

84

84

0

30

38

143

65

86

83

84

150

160

147

3

181

78

78

0

122

38

129

144

116

13

64

54

10

97

32

191

176

176

0

70

66

133

24

78

95

120

16

148

13

67

197

143

143

0

180

145

6

84

55

185

7

77

174

25

41

199

174

174

0

105

162

83

129

100

104

106

36

10

83

176

211

165

165

0

192

78

98

178

173

190

77

96

101

58

86

211

182

182

0

155

11

37

210

111

113

126

10

199

119

24

223

69

69

0

41

45

55

211

154

104

52

8

140

215

115

229

44

44

0

120

127

35

174

225

143

172

7

161

17

73

229

209

209

0

219

104

7

38

62

94

32

167

209

89

92

233

33

33

0

34

13

124

218

79

170

138

23

22

145

55

241

94

94

0

200

219

44

40

139

133

194

33

79

216

98

257

48

48

0

35

193

232

136

111

42

246

8

240

108

153

263

79

79

0

228

138

101

146

179

235

11

154

235

65

25

269

171

171

0

113

237

101

88

201

166

6

182

69

230

91

269

180

180

0

74

218

219

202

244

259

140

164

82

64

245

269

207

207

0

77

22

243

185

188

90

167

129

189

99

31

281

20

20

0

127

19

267

238

270

101

2

147

168

255

241

283

147

147

0

26

218

169

270

98

171

149

208

28

73

159

293

91

91

0

39

206

171

105

26

97

289

219

38

212

97

307

40

40

0

65

32

263

223

198

59

173

201

300

51

156

313

104

104

0

218

120

107

74

51

239

108

42

125

305

227

313

213

213

0

58

53

223

63

66

120

238

69

293

236

74

331

18

18

0

14

185

36

326

146

77

236

117

314

217

208

331

71

71

0

147

280

26

271

249

28

266

190

81

72

272

331

324

324

0

173

41

188

39

158

186

190

192

261

44

66

347

75

75

0

27

292

130

202

322

311

308

91

124

68

45

347

156

156

0

204

298

48

283

227

292

96

238

158

288

171

349

223

223

0

164

173

80

185

86

312

39

96

135

319

200

349

317

317

0

103

38

68

134

72

47

119

213

260

143

75

353

14

14

0

337

259

116

221

334

144

288

263

294

249

149

353

196

196

0

258

218

347

348

162

301

246

26

14

219

298

359

257

257

0

7

62

98

138

61

136

346

355

202

95

59

359

331

331

0

220

114

233

245

242

343

204

171

157

105

184

367

159

159

0

8

121

119

104

131

157

155

171

361

260

334

367

205

205

0

357

199

82

254

320

156

227

196

115

135

257

373

242

242

0

85

355

365

313

261

262

275

8

304

76

16

379

174

174

0

198

72

194

145

375

164

136

90

120

97

280

397

175

175

0

184

332

45

282

293

362

7

18

355

77

396

401

280

280

0

111

66

332

351

28

254

130

160

12

373

20

419

369

369

0

210

182

164

36

27

57

284

65

109

78

307

421

251

251

0

391

78

129

155

80

102

361

409

247

41

368

433

349

349

0

226

95

120

215

8

94

113

137

206

50

182

439

194

194

0

171

199

141

134

421

15

420

120

321

21

35

449

210

210

0

241

129

137

282

393

314

294

83

246

101

212

461

52

52

0

206

196

154

265

320

98

291

416

232

273

451

463

255

255

0

177

75

422

113

430

83

351

42

169

375

265

463

345

345

0

40

331

112

286

24

426

256

175

307

436

370

487

10

10

0

329

462

103

341

149

54

64

287

349

100

311

487

100

100

0

249

0

258

336

428

281

36

237

54

415

322

487

175

175

0

427

44

471

72

106

273

475

187

406

475

207

487

307

307

0

424

391

356

98

429

194

148

479

236

420

3

499

346

346

0

460

279

269

135

282

230

492

301

219

312

420

509

93

93

0

263

29

65

137

86

45

367

277

294

481

493

509

250

250

0

362

34

224

121

181

23

163

373

211

115

476

521

308

308

0

234

382

126

54

143

270

374

143

129

105

504

523

241

241

0

177

363

517

57

483

214

449

291

455

515

365

547

427

427

0

80

35

1

9

519

12

248

135

52

437

525

557

430

430

0

216

407

507

519

454

443

482

394

423

307

4

563

486

486

0

95

227

453

37

221

314

138

122

430

552

234

571

262

262

0

465

32

170

218

521

229

319

188

359

219

231

571

422

422

0

235

88

536

559

257

229

106

540

414

430

452

571

516

516

0

432

328

306

112

386

407

316

168

298

190

9

577

427

427

0

173

365

178

420

563

350

442

230

281

568

135

599

559

559

0

22

516

267

499

67

485

573

452

454

565

338

599

588

588

0

589

354

135

381

250

166

129

215

282

372

246

601

405

405

0

206

198

25

241

258

187

133

391

525

404

295

607

162

162

0

28

95

427

332

475

143

10

374

525

53

61

607

302

302

0

298

328

238

332

253

427

365

482

386

364

1

617

556

556

0

295

98

413

185

346

272

288

5

439

33

555

619

286

286

0

412

73

19

138

443

323

552

101

519

576

4

631

69

69

0

436

306

380

15

431

309

438

213

508

462

250

631

534

534

0

166

492

272

428

130

153

213

346

378

261

273

641

340

340

0

262

69

543

35

618

472

390

519

613

20

474

647

56

56

0

140

27

415

35

514

425

378

583

89

458

253

647

526

526

0

424

398

143

435

400

187

561

421

578

392

523

653

84

84

0

256

398

21

227

576

63

105

604

360

551

337

653

120

120

0

380

432

266

387

96

541

640

138

311

454

151

653

197

197

0

488

104

542

530

73

535

89

26

154

125

304

653

287

287

0

242

101

139

90

595

75

26

265

559

647

617

653

410

410

0

550

517

360

607

363

295

451

73

367

504

652

659

503

503

0

332

163

234

104

573

123

3

637

498

72

46

661

184

184

0

24

88

510

39

522

472

487

446

32

600

5

673

22

22

0

131

423

461

316

512

460

298

0

549

192

316

673

484

484

0

380

449

457

280

57

208

610

668

555

341

244

701

375

375

0

608

72

366

153

324

202

483

404

383

105

355

727

92

92

0

148

465

314

508

11

31

88

13

164

581

357

739

168

168

0

338

187

646

243

602

64

62

195

681

478

230

743

467

467

0

301

187

637

366

614

176

85

187

552

288

193

761

675

675

0

240

616

172

161

269

43

462

481

759

669

550

769

392

392

0

749

733

564

221

316

264

293

466

340

696

542

773

392

392

0

575

155

542

561

413

446

173

546

81

245

375

797

440

440

0

651

772

713

596

647

154

69

511

416

563

228

797

446

446

0

133

654

744

149

541

441

195

222

529

417

1

809

207

207

0

136

315

270

374

671

514

533

322

556

253

603

823

393

393

0

546

57

76

596

160

414

133

740

627

385

297

827

314

314

0

587

431

569

179

608

385

0

118

506

728

509

827

381

381

0

471

730

400

773

607

299

760

280

656

178

480

829

46

46

0

416

576

9

262

755

19

576

659

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