examples of ranges of consecutive integers for Erdős-Woods numbers

examples of ranges of consecutive integers for Erdős-Woods numbers

The most famous example of 16 as an Erdős-Woods number (

http://planetmath.org/ErdHosWoodsNumber

) is the range of 16 consecutive

integers

starting with 2184.

Another

n

for

k

=

16

is 2044224, which we obtained by multiplying 2184 by 936. The factorization is

2044224

=

2

6

×

3

3

×

7

×

13

2

, while

2044224

+

16

=

2044240

=

2

4

×

5

×

11

×

23

×

101

. The table of factorizations

2044225

5

2

×

81769

2044226

2

×

1022113

2044227

3

×

681409

2044228

2

2

×

511057

2044229

11

×

19

×

9781

2044230

2

×

3

×

5

×

68141

2044231

7

2

×

41719

2044232

2

3

×

59

×

61

×

71

2044233

3

2

×

17

×

31

×

431

2044234

2

×

1009

×

1013

2044235

5

×

107

×

3821

2044236

2

2

×

3

×

170353

2044237

13

×

67

×

2347

2044238

2

×

7

×

151

×

967

2044239

3

×

29

×

23497

shows that each of the numbers in this range shares at least one factor with one if not both of the numbers capping the range.

Next we have a slightly longer example, this one for

k

=

34

. The smallest

matching

n

is 47563752566, a

squarefree

number with a factorization of

2

×

11

×

17

×

23

×

41

×

157

×

859

. The number capping the end of the range is the decidedly non-squarefree 47563752600, with a factorization of

2

3

×

3

2

×

5

2

×

7

×

13

×

17

×

19

×

29

×

31

. While the size of these numbers forbids verification on your typical pocket calculator, these numbers are well within the reach of a

Javascript

implementation of

trial division

. Here we could be tempted to omit the

even numbers

, as they obviously share 2 as a

prime factor

with the range start and the range end, as well as

multiples

of 3 or 5 as they thus share factors with the range end. But, on the hope that it turns out to be at least a little bit instructive, the factorizations of all the numbers in our chosen range is given.

47563752567

3

×

3719

×

4263131

47563752568

2

3

×

71

×

199

×

420799

47563752569

31

×

163

×

9412973

47563752570

2

×

3

×

5

×

1585458419

47563752571

29

×

12941

×

126739

47563752572

2

2

×

7

2

×

242672207

47563752573

3

2

×

4657

×

1134821

47563752574

2

×

13

×

823

×

991

×

2243

47563752575

5

2

×

31769

×

59887

47563752576

2

7

×

3

×

349

×

354911

47563752577

11

×

397

×

593

×

18367

47563752578

2

×

173

×

137467493

47563752579

3

×

7

×

3257

×

695407

47563752580

2

2

×

5

×

83

×

617

×

46439

47563752581

19

×

2503355399

47563752582

2

×

3

4

×

53

×

59

×

93893

47563752583

17

×

43

×

5171

×

12583

47563752584

2

3

×

149

×

39902477

47563752585

3

×

5

×

67

×

47327117

47563752586

2

×

7

×

3397410899

47563752587

13

2

×

281442323

47563752588

2

2

×

3

×

11

×

4513

×

79843

47563752589

23

×

61

×

151

×

224513

47563752590

2

×

5

×

4756375259

47563752591

3

2

×

5284861399

47563752592

2

4

×

47

×

63249671

47563752593

7

×

6794821799

47563752594

2

×

3

×

7927292099

47563752595

5

×

32503

×

292673

47563752596

2

2

×

11890938149

47563752597

3

×

15854584199

47563752598

2

×

23781876299

47563752599

11

×

37

×

127

×

373

×

2467

Title

examples of ranges of consecutive integers for Erdős-Woods numbers

Canonical name

ExamplesOfRangesOfConsecutiveIntegersForErdHosWoodsNumbers

Date of creation

2013-03-22 17:38:16

Last modified on

2013-03-22 17:38:16

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 11A05

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