Searching for large Sierpinski numbers

overlaps

flip cycles

In an earlier article, I introduced a method for generating numbers which have a good chance of having a Nash weight of less than 100. A side-effect of this is of course the locating of Sierpinski numbers themselves, which have a Nash weight of zero. In fact the method may be modified specifically to find Sierpinski numbers, by ignoring k as soon as any value of n between 1 and 1000 survives the trial division, although in practice it is convenient to preserve the possibility of one or two exceptions. It takes only a few minutes to run this for all k up to one million, and it produces a list of 11 survivors, all of which, when run through the Nash sieve, are identified as Sierpinski numbers.

It is important to point out that for each k, the exponent n must be run to 1000. For example, if the modified algorithm is run for odd k between 70000 and 100000, and n only up to 100, then instead of leaving one possible candidate, there are 16 to be considered. Some of these, to be fair, have very low Nash weights, but only one (78557) has zero Nash weight.

It is possible that additional Sierpinski numbers are not identified because either

a) the Sierpinski number is divisible by 3, 5 or 7

or

b) a member of the covering set is greater than 6400 (the sieve limit).

As will become apparent, the dangers posed by condition a) are removed by the nature of certain relationships governing Sierpinski numbers. Also, there is a margin of error in the search algorithm that will cope with covering sets with one or two large primes.

The following list gives all Sierpinski numbers found using this method for k < 107. As a comparison, the algorithm was run for all k < 5.106 without the restriction that k is not divisible by 3, 5, or 7, and no other Sierpinski numbers were located. I am therefore confident that the number of Sierpinski numbers not identified in the range of k searched is very small and may, in fact, be zero.

The covering set for a Sierpinski number k is a subset of the divisors of 2s - 1 for some minimal value s. This number is given in the second column and is called the Sierpinski modulus for k. Covering sets themselves are letter-coded in the fourth column, as follows :

code

modulus

covering set

A

24

{3, 5, 7, 13, 17, 241}

B

36

{3, 5, 7, 13, 19, 37, 73}

C

48

{3, 5, 7, 13, 17, 97, 257}

D

36

{3, 5, 7, 13, 19, 73, 109}

E

36

{3, 5, 7, 13, 19, 37, 109}

F

36

{3, 5, 7, 13, 37, 73, 109}

G

72

{3, 5, 7, 13, 17, 19, 109, 433}

It is obvious that if k is a Sierpinski number and P is the product of the primes in the covering set for k, then all the numbers generated by the iteration k ® (2k+P) mod 2P are also Sierpinski numbers with the same covering set. This iterative process repeats with period equal to the Sierpinski modulus s of k and is known as the Keller iteration. The sequence of s repeating numbers so generated is called a Keller cycle. In the fifth column below we attempt to indicate this using the following code: proceeding sequentially, if a value of k has not been previously identified as being a member of a particular cycle, then it is given the code N.0 where N is a number that starts at 1 and is incremented every time a new cycle is identified. The Keller cycle is then generated and a member of the cycle is labelled N.i if it is the ith iteration of the cycle. We proceed in this manner until all the values of k are labelled. Note that N.0 = N.s, where s is the Sierpinski modulus.

For interest, the third column contains either the word "prime" if k is prime, or the prime factors of k otherwise. It is hoped that a pattern may emerge regarding divisibility properties.

The Sierpinski numbers are :

k

modulus

factorisation

covering set

label

78557

36

17.4621

B

1.0

271129

24

prime

A

2.0

271577

24

59.4603

A

3.0

322523

36

prime

F

4.0

327739

48

prime

C

5.0

482719

24

prime

A

3.17

575041

24

29.79.251

A

2.6

603713

24

11.71.773

A

3.6

903983

24

149.6067

A

2.13

934909

36

prime

D

6.0

965431

24

227.4253

A

2.18

1259779

36

23.54773

D

6.24

1290677

36

137.9421

E

7.0

1518781

24

11.138071

A

3.15

1624097

24

887.1831

A

2.11

1639459

24

prime

A

2.16

1777613

72

29.61297

G

8.0

2131043

24

prime

A

2.21

2131099

24

prime

A

3.21

2191531

36

103.21277

B

1.31

2510177

36

167.15031

B

9.0

2541601

24

43.59107

A

3.11

2576089

36

prime

B

9.21

2931767

24

859.3413

A

2.23

2931991

24

37.109.727

A

3.23

3083723

24

443.6961

A

2.5

3098059

24

prime

A

3.5

3555593

24

23.154591

A

3.14

3608251

24

prime

A

2.10

4067003

24

37.109919

A

3.10

4095859

36

41.283.353

E

7.29

4573999

24

prime

A

3.13

5455789

48

59.89.1039

C

10.0

5841947

36

271.21557

E

11.0

6134663

24

19.322877

A

2.1

6135559

24

29.211571

A

3.1

6557843

24

37.177239

A

3.18

6676921

24

197.33893

A

2.2

6678713

24

prime

A

3.2

6742487

24

1327.5081

A

2.7

6799831

24

prime

A

3.7

6828631

36

23.337.881

D

12.0

7134623

36

232.13487

B

9.14

7158107

36

11.191.3407

E

7.20

7400371

24

11.101.6661

A

2.14

7523267

24

29.59.4397

A

2.19

7523281

24

prime

A

3.19

7696009

36

83.92723

B

1.25

7761437

24

prime

A

2.3

7765021

24

11.113.6247

A

3.3

7892569

24

31.47.5417

A

2.8

8007257

24

503.15919

A

3.8

8184977

36

prime

B

13.0

8629967

24

41.210487

A

3.16

8840599

24

prime

A

2.12

8871323

24

401.22123

A

2.17

8879993

48

prime

C

10.33

8959163

36

prime

E

14.0

9043831

48

283.31957

C

5.42

9044629

36

112.17.4397

E

7.13

9208337

24

prime

A

2.15

9252323

36

prime

E

11.14

9454129

24

37.255517

A

2.20

9454157

24

73.129509

A

3.20

9854491

24

163.60457

A

2.22

9854603

24

112.23.3541

A

3.22

9930469

24

prime

A

2.4

9933857

36

312.10337

E

7.24

9937637

24

prime

A

3.4

The counts for each covering set are as follows:

Total

A

B

C

D

E

F

G

69

45

7

4

3

8

1

1

In the above list, there are several obvious pairs, with differences, as follows :

271129 + 26.7 = 271577

2131043 + 23.7 = 2131099

2931767 + 25.7 = 2931991

6134663 + 27.7 = 6135559

6676921 + 28.7 = 6678713

7523267 + 2.7 = 7523281

7761437 + 29.7 = 7765021

9454129 + 22.7 = 9454157

9854491 + 24.7 = 9854603

where I have laid these out so that the striking similarity of the differences is apparent. Two other possible pairings, {8871323, 8879993} and {9043831, 9044629}, do not provide anything of interest, and the "closeness" of these pairs may be put down to co-incidence. There is one other grouping of interest, that is, the last three. The two consecutive differences do not indicate anything of significance. However, the following may seem rather striking :

9930469 + 210.7 = 9937637

In other words, the difference 2m.7 occurs once and only once for each m. It comes as no surprise to find that all these pairs have the same Sierpinski modulus and covering set, whereas the two other pairs mentioned do not. Additional investigation of differences between non-adjacent Sierpinski numbers reveals the following :

3083723 + 211.7 = 3098059

This effect is a direct result of the cyclic generating procedure, since if k and k' have the same covering set and we let x = k' - k, then the differences between successive iterations in the Keller cycle are just 2x, 4x, etc. In the above example, we may take the closest pairing, that is, k = 7523267 and k' = 7523281 as the starting values of the cycles.

Obviously, x = k' - k is even. The above example has x = 2*7. It would be interesting to find examples with x = 2*3 or 2*5, although at present no other pair of known Sierpinski numbers comes anywhere near as close as this.

We shall now continue the list to 2*107. It is obvious that the alternative iteration k ® (k+2P) provides an infinite sequence of Sierpinski numbers. However, these have identical properties to the associated base values and will be labelled with an asterisk. Applying the Keller iteration to any one of these generated values will re-create the Keller cycle for the base value. A Keller cycle is therefore uniquely defined by a base value N.0 and a covering set.

10192733

24

prime

A

2.9

10275229

36

31.461.719

B

9.5

10306187

36

prime

E

15.0

10391933

36

prime

B

1.10

10422109

24

47.221747

A

3.9

10675607

24

1297.8231

A

3.12

11000303

36

43.47.5443

E

15.22

11201161

36

23.487007

B

13.31

11206501

48

prime

H

16.0

11455939

24

11.1041449

A

*

11456387

24

prime

A

*

11667529

24

prime

A

*

11759851

24

661.17791

A

*

11788523

24

prime

A

*

11822359

36

prime

D

17.0

12088793

24

prime

A

*

12150241

24

521.23321

A

*

12151397

36

149.81553

B or D or J

9.8 or 12.5 or 18.0

12384413

36

337.36749

B

9.26

12413281

36

17.191.3823

F

19.0

12703591

24

149.85259

A

*

12756019

36

prime

B

20.0

12808907

24

19.23.29311

A

*

12824269

24

67.277.691

A

*

13065289

36

89.146801

B

13.13

13085029

36

43.304303

B

1.13

13315853

24

43.309671

A

*

13315909

24

prime

A

*

13726411

24

prime

A

*

14116577

24

prime

A

*

14116801

24

1873.7537

A

*

14268533

24

23.41.15131

A

*

14282869

24

397.35977

A

*

14740403

24

prime

A

*

14793061

24

prime

A

*

15168739

36

3253.4663

B

19.20

15251813

24

19.67.11981

A

*

15273751

48

prime

I

21.0

15285707

36

prime

D

12.13

15598231

36

112.17.7583

E

11.7

15758809

24

11.19.75401

A

*

16010419

36

101.158519

E

15.5

16391273

36

71.230863

B

19.9

16625747

36

prime

E

7.8

16907749

36

73.231613

E

11.25

16921847

36

prime

F

18.13

17220887

36

3709.4643

D

22.0

17319473

24

prime

A

*

17320369

24

11.1574579

A

*

17742653

24

prime

A

*

17861731

24

23.672.173

A

*

17863523

24

601.29723

A

*

17927297

24

prime

A

*

17984641

24

prime

A

*

18068693

36

prime

E

11.30

18140153

36

prime

B

9.34

18156631

36

prime

B

9.19

18585181

24

prime

A

*

18708077

24

2707.6911

A

*

18708091

24

prime

A

*

18946247

24

239.79273

A

*

18949831

24

41.462191

A

*

19077379

24

67.284737

A

*

19192067

24

prime

A

*

19428919

36

prime

E

15.13

19436611

36

197.98663

B

1.23

19558853

36

971.20143

B

13.34

19814777

24

19.47.22189

A

*

where we have the additional coded covering sets :

code

modulus

covering set

H

48

{3, 5, 7, 17, 97, 241, 257}

I

48

{3, 5, 7, 13, 17, 257, 673}

J

36

{3, 5, 7, 13, 19, 73, 4033}

The algorithm actually located one value, k = 13965257, which had a zero intermediate weighting but a non-zero Nash weight and is therefore not a Sierpinski number.

As has been indicated above, it is entirely possible for a particular Sierpinski number to have more than one covering set. We shall call such numbers "overlapping" Sierpinski numbers, or "overlaps". The set J is not a true covering set since 4033 = 37*109 is not a prime. However, it is included in order to label these overlaps conveniently.

For covering set A, we have 2P(A) = 11184810. The only two base values with this covering set that are less than this bound are 271129 and 271577. Since the search has now passed this value of 2P, all other Sierpinski numbers with this covering set can be generated from one of these two by using either the Keller iteration or the additive iteration. This idea stretches to all Sierpinski numbers, that is, for a given covering set and modulus, all Sierpinski numbers k such that k > 2P have an associated base value k' such that k' < 2P. Thus there is a finite number of base values, and therefore also of Keller cycles, for each finite covering set.

None of the above Sierpinski numbers is divisible by 13. However, we cannot assume that this is a universal property. The reasoning for this is as follows: the number k = 11206501 has covering set H, as can be seen above, which does not contain 13. The product P(H) is approximately 1010 and the Sierpinski modulus is 48. Applying the Keller iteration and ignoring any number greater that 231 (which is the limit of the Nash-Jobling implementation of the Nash sieve), we are left with the following :

iteration #

Sierpinski number

factorisation

Sierpinski modulus

10

751375159

19.prime

48

14

1297920679

11.31.317.12007

24

38

701684269

11.13.19.101.2557

48

42

502866439

prime

48

Thus 701684269 is a Sierpinski number which is divisible by 13. A bonus here is that this value is 20 times smaller than the expected theoretical value.

Iteration 14 above is unusual. Although the original covering set still covers this value of k, the prime 13 has the same effect as 97 and 257 combined. We shall return to this idea shortly.

The 22 Keller cycles identified with smallest values less than 2*107 are divided amongst covering sets as follows :

A

271129 (2)

271577 (3)

 

 

B

78557 (1)

2510177 (9)

8184977 (13)

12756019 (20)

C

327739 (5)

5455789 (10)

 

 

D

934909 (6)

6828631 (12)

11822359 (17)

17220887 (22)

E

1290677 (7)

5841947 (11)

8959163 (14)

10306187 (15)

F

322523 (4)

12413281 (19)

 

 

G

1777613 (8)

 

 

 

H

11206501 (16)

 

 

 

I

15273751 (21)

 

 

 

J

12151397 (18)

 

 

 

where the values in brackets are the labels given to the Sierpinski number in the earlier lists.

There are 7 Sierpinski numbers that overlap between cycles with different covering sets. These are :

12151397 - shared by cycles 9 (B) and 12 (D)

45181667 - shared by cycles 13 (B) and 6 (D)

68468753 - shared by cycles 1 (B) and 17 (D)

69169397 - shared by cycles 21 (D) and 11 (E)

71307347 - shared by cycles 1 (B) and 17 (D)

182479909 - shared by cycles 12 (D) and 14 (E)

392581699 - shared by cycles 6 (D) and 18 (F)

The four overlaps involving covering sets B and D are similar in that the primes 37 and 109 are directly interchangeable since they cover the same exponents, that is, have the same Nash congruence. Since these two primes both have exponent 36 to base 2, there is a 35 to 1 chance that a Sierpinski number covered by B or D is covered by both B and D. Any overlap between B and D is automatically covered also by J since 4033 = 37*109.

The two overlaps involving covering sets D and E are more complicated in that the subsets {3,5,73) of D and {3,5,37} of E are interchangeable but no smaller subsets are interchangeable. This is because the Nash congruence for 37 is 27 mod 36 and for 73 is 0 mod 9. Thus 73 covers everything covered by 37 but the reverse is not the case. However, 0, 9 and 18 mod 36 are covered by the primes 3 and 5 so the net effect of the triplets is the same. Similarly, with the overlap involving covering sets D and F, the subsets {5,19} and {5,37} are interchangeable since 19 covers both 4 and 22 mod 36 whereas 37 covers only 22 mod 36, but the prime 5 covers all numbers divisible by 4. In the event of an overlap between D and E or D and F, the set D is considered to be the better covering set since the exponent of the exceptional prime is lower in each case.

Other overlaps exist, for instance the one mentioned previously with respect to the search for a Sierpinski number divisible by 13, in which the overlap involves covering sets of different moduli. A complete picture of overlaps involving covering set A is missing since the product of primes is so small that it requires generated values to match those in Keller cycles of other covering sets and only base values where compared above.

Continuing the search to 5*107 reveals 3 more covering sets, 7 more Keller cycles and 8 more overlaps between different Keller cycles. Ignoring those Sierpinski numbers with modulus 24 (as they are all generated values), the list continues :

20189993

36

967.20879

E

11.10

20312899

36

43.472393

B

13.29

20778931

36

71.292661

B

13.11

21610427

36

prime

B

20.7

21823667

48

937.23291

C

5.19

22024609

72

1303.16093

K

23.0

22047647

36

1481.14887

B

9.32

23277113

36

79.294647

D

6.13

23487497

36

11.2135227

E

15.8

24885199

48

113.191.1153

L

24.0

25614893

36

23.311.3581

E

14.24

25763447

48

983.26209

C

10.15

25912463

36

47.479.1151

D

6.33

26471633

36

672.5897

E

7.34

27160741

36

373.72817

B

9.11

27862127

36

29.960763

F

25.0

28410121

36

113.251417

E

7.11

29024869

36

67.433207

B

9.29

29095681

36

839.34679

F

25.27

29949559

48

prime

C

5.32

30375901

36

prime

E

11.23

30423259

36

prime

B

20.28

30666137

36

29.47.149.151

E

11.28

31997717

36

prime

B

20.3

32548519

36

23.53.26701

B

20.24

32552687

36

67.289.1249

E

14.22

32971909

36

311.106019

D

6.8

33234767

36

prime

B

20.31

33485483

36

prime

B

13.6

33742939

36

283.119233

E

11.21

34167691

36

929.36779

B

20.14

34471877

36

11.1493.2099

B

13.16

34629797

36

193.179429

B

1.16

34636643

36

23.29.51929

B

13.22

34689511

36

prime

D

12.28

35430841

36

127.227.1229

F

4.21

36029731

72

71.507461

G

26.0

36120983

36

31.1165193

B

1.30

36234799

36

3307.10957

E

14.31

37158601

48

prime

M

27.0

38206517

36

prime

D

12.17

38222131

36

349.109519

E

15.27

38257411

36

prime

D

22.19

38592529

36

4937.7817

B

9.13

38750753

36

2971.13043

E

14.4

38942027

36

1049.37123

E

14.14

39953689

36

17.2350217

E

11.33

40118209

36

prime

E

7.17

40343341

36

prime

E

14.33

40511719

36

43.83.11351

E

15.33

41134369

48

prime

L

28.0

41403227

36

3533.11719

B

20.35

42609587

36

prime

B

20.19

43441313

36

211.205883

E

11.18

43925747

36

43.107.9547

F

25.4

44091199

36

prime

E

14.11

44103533

36

4751.9283

B

9.18

44743523

36

11.4067593

B

1.22

45181667

36

prime

B or D or J

13.28 or 6.19 or 29.0

45414683

36

61.744503

B

13.10

45830431

36

31.491.3011

B

20.6

46049041

36

prime

B

9.31

46337843

36

4327.10709

D

22.14

47635073

36

prime

F

19.23

48292669

36

151.319819

E

15.29

48339497

36

107.451771

D

17.15

The new covering sets are :

code

modulus

covering set

K

72

{3, 5, 7, 13, 17, 19, 37, 433}

L

48

{3, 5, 7, 13, 17, 97, 673}

M

48

{3, 5, 13, 17, 97, 241, 257}

The last of these does not have the prime 7 as a member. Although the value k = 37158601 is a prime, the Keller cycle associated with it contains seven members that are divisible by 7, the lowest of which is k = 1905955429. In fact, another member of the cycle, k = 16352358743, is divisible by 72. The direct search algorithm used to locate the Sierpinski numbers listed above ignores numbers divisible by 7, but the nature of the Keller iteration ensures that most of members of the Keller cycle are not, and so once one of these is found, any smaller member of the cycle can be retrieved. An example of this will eventually be forthcoming.

The additional overlaps found are :

154337567 - shared by cycles 17 (D) and 25 (F)

226521259 - shared by cycles 6 (D) and 29 (J)

236281883 - shared by cycles 6 (D) and 29 (J)

282777829 - shared by cycles 12 (D) and 18 (J)

300943667 - shared by cycles 17 (D) and 25 (F)

343302301 - shared by cycles 17 (D) and 25 (F)

1300668683 - shared by cycles 8 (G) and 23 (K)

4213021013 - shared by cycles 8 (G) and 23 (K)

The overlaps shared by sets D and J also occur as generated values for set B. The overlaps involving G and K are also based around the 37-109 interchangeability.

Totals for Sierpinski numbers by covering set are :

Total

A

B

C

D

E

F

G

H

I

J

K

L

M

332

213

44

7

14

36

8

2

1

1

2

1

2

1

where the two overlaps that fall into the search range have been assigned to covering set J.

We now complete the search to 108. As before, we shall ignore the generated values associated with covering set A.

50236847

36

5779.8693

B

20.27

50273851

36

prime

F

4.33

50407157

36

73.690509

E

14.6

50835497

36

23.139.15901

E

15.16

51172253

36

11.97.199.241

E

14.16

51299477

36

prime

B

20.23

51612259

72

149.346391

N

30.0

51642601

36

prime

B

20.30

51767959

36

3109.16651

B

13.5

51889823

72

2347.22109

P

31.0

52109063

36

17.257.11927

B

20.13

52343539

36

131.463.863

B

13.21

53085709

36

79.671971

B

1.29

54345857

36

prime

E

7.28

55218901

36

71.777731

E

11.35

55726831

36

412.33151

B

20.34

55876981

36

1373.40697

E

7.19

56191673

72

79.711287

P

31.48

56330011

36

173.325607

B

20.18

56517767

36

7013.8059

F

4.30

56777509

36

127.447067

E

14.35

56924089

36

113.503753

E

11.13

57396979

36

2557.22447

B

1.21

57410477

36

127.251.1801

D

6.11

57451021

36

389.147689

E

15.35

57616051

36

1021.56431

B

13.27

57732559

36

83.695573

B

13.9

57798079

36

17.3399887

E

15.21

57816799

36

53.1090883

F

32.0

57940433

36

821.70573

B

20.5

59198569

72

prime

K

33.0

59929127

36

479.125113

F

4.14

60097043

36

5653.10631

E

11.6

60143641

36

prime

B

20.26

60303137

36

prime

E

15.4

60610801

36

prime

E

7.7

60666107

36

19.3192953

F

32.3

60909197

36

89.684373

B

13.4

61079749

36

167.365747

B

20.12

61196987

36

prime

B

13.20

62012387

36

131.473377

E

15.12

62888633

36

prime

B

20.33

63190223

36

23.2747401

B

20.17

63662611

36

5381.11831

F

25.31

63676073

36

prime

D

17.13

63723707

36

71.897517

B

1.20

63833243

36

4951.12893

B

13.26

63891497

36

887.72031

B

13.8

65623711

36

prime

B

13.19

66620329

36

2267.29387

B

20.16

66887071

36

929.71999

B or F

1.19 or 32.14

66941839

36

prime

B

13.25

67510217

36

103.655439

D

17.11

67800683

48

19.3568457

L

24.25

67837073

36

2749.24677

B

13.18

68468753

36

179.382507

B or D or J

1.18 or 17.9 or 34.0

68496137

36

prime

B

13.24

68574271

36

prime

E

14.21

68708387

36

11.61.102397

D

17.7

69169397

36

11.227.27701

E or D

11.20 or 22.4

69265069

36

67.1033807

F

19.30

70207549

36

2557.27457

B

1.1

70312793

36

601.116993

D

22.6

70364663

36

prime

B

1.2

70415327

36

41.1717447

E

14.30

70678891

36

101.699791

B

1.3

71151293

36

31.163.14081

D

12.31

71307347

36

31.313.7349

B or D or J

1.4 or 17.31 or 34.22

71408993

36

17.4200529

E

15.26

71476051

36

137.521723

D

22.31

71768941

36

151.461.1031

E

14.13

72553787

36

587.123601

E

15.32

72564259

36

1223.59333

B

1.5

74343527

36

5591.13297

E

14.10

74433497

36

17.23.190367

B

1.32

74886377

36

17.41.107441

D

22.8

75037639

48

31.2420569

C

5.16

75070789

36

557.134777

B

9.1

75078083

36

2699.27817

B

1.6

75202613

36

prime

B

9.22

77564731

36

71.1092461

F

4.17

77807131

48

prime

L

24.6

78240377

36

prime

D

12.33

78816559

36

prime

B

1.33

78864593

36

643.122651

D

17.33

79539409

36

523.152083

D

22.33

80091143

36

11.59.123407

B

9.2

80105731

36

5737.13963

B

1.7

80354791

36

11.7304981

B

9.23

81196967

36

31.2619257

E

15.20

81722987

36

179.456553

D

12.25

82346449

36

89.925241

E

11.5

83304121

36

1063.78367

E

15.11

83460571

36

71.367.3203

F

32.6

84319681

36

173.487397

B

9.15

85442453

36

241.354533

B

1.26

86420389

36

11.7856399

B

13.1

86585063

36

17.317.16067

E

14.20

87376127

36

127.397.1733

F

32.23

87505591

36

31.2822761

E

14.29

87582683

36

41.2136163

B

1.34

88574821

36

prime

E

15.31

89469691

36

17.5262923

E

14.9

90131851

36

113.797627

B

9.3

90161027

36

11.59.138923

B

1.8

90600893

36

prime

B

9.6

90659147

36

17.317.16823

B

9.24

90834301

36

2437.37273

B

1.11

92452757

36

31.173.17239

B

13.32

92732027

48

19.37.131909

C

5.11

92896411

36

347.267713

E

15.19

93180713

36

4793.19441

D

22.10

94353229

36

prime

B

9.9

94741307

36

5471.17317

D

17.27

94751851

36

1109.85439

D

22.27

94819261

36

1153.82237

B

9.27

95562473

36

103.927791

B

20.1

95590459

36

1163.82193

E

14.19

96050723

36

7487.12829

E

14.28

96181013

36

1151.83563

B

13.14

96220493

36

17.53.269.397

B

1.14

97032773

36

41.2366653

E

14.8

98588927

36

499.197573

B

9.16

98746133

36

2011.49103

E

15.18

98915393

48

37.1097.2437

L

24.21

99287341

36

6451.15391

D

12.20

99694493

36

41.2431573

D

22.22

The new covering sets are :

code

modulus

covering set

N

72

{3, 5, 7, 17, 19, 37, 109, 241}

P

72

{3, 5, 7, 17, 19, 37, 73, 241}

Additional overlaps found including new cycles are :

66887071 - shared by cycles 1 (B) and 32 (F)

114921271 - shared by cycles 7 (E) and 32 (F)

122311103 - shared by cycles 22 (D) and 32 (F)

133228283 - shared by cycles 7 (E) and 32 (F)

192413177 - shared by cycles 22 (D) and 32 (F)

In the B & F overlap, the subsets {5,19} and {5,109} are interchangeable. The overlaps involving sets E and F are as a result of a {3,19} and {3,73} interchange. Since 73 has the smaller exponent, the primary covering set is F.

From the data obtained, it is increasingly obvious that for each different covering set there will be an even number of distinct Keller cycles. This conjecture is definitely true of set A, since the 2P barrier has been reached. Continuing the search to 1.5*108 would pass the 2P barrier for set B. At the moment, there are 4 Keller cycles associated with set B, and it is unlikely that any more will be found. There are also 4 distinct Keller cycles currently known for sets D, E and F, with no more expected. Justification for the conjecture is as follows.

Suppose k is a Sierpinski number with covering set P = {p1, p2, } and modulus m and such that k < 2P. Let ei be the exponent of pi to base 2, (so that lcm(e1, e2, ) = m) and let the Nash congruences associated with P be (pi, ri). The j-th step of the Keller iteration k ® (2k+P) mod 2P changes the Nash congruences to (pi, ri - j), in effect, sliding the divisibility properties one place to the left. Since the ri are all reduced modulo ei , the m-th step re-aligns all the congruences and we arrive back at k. Thus we have the Keller cycle relating m different Sierpinski numbers. However, rather than simply shifting the congruences by 1, we may consider flipping the congruences about the midpoint m/2. In this case, the new congruences are (pi, m+1- ri) associated with a mirror-image or flip Sierpinski number k'. The second term here can be replaced by ei+1- ri since ei divides m. Repeating this flip returns the original values. We have k.+1 0 mod pi and so k - mod pi . Replacing the exponent, we have k' - mod pi . We then have a set of simultaneous congruences that can be solved to obtain k' < 2P, k odd, which always exists by the Chinese Remainder Theorem. We can then generate a flip cycle using the Keller iteration.

As an example, k = 271129 has covering set A, modulus 24, and Nash congruences obtained using psieve are (3,1) (5,0) (7,2) (13,6) (17,6) and (241,10) and Keller label 2.0. Since the primes involved have exponents 2, 4, 3, 12, 8 and 24 respectively, the flipped Nash congruences are (3,0) (5,1) (7,2) (13,7) (17,3) and (241,15), leading to the set of congruences k' 2 mod 3, k' 2 mod 5, k' 5 mod 7, k' 7 mod 13, k' 2 mod 17, k' 211 mod 241. The smallest solution of this is k' = 271577, which is the smallest value in the only other Keller cycle for set A (Keller label 3.0). Thus there is a direct match between members of the two Keller cycles, that is, the j-th Keller iterations are flips of each other for all j < 24.

A more complicated example is provided by k = 78557 (Keller label 1.0), the smallest Sierpinski number. This has covering set B, modulus 36, and Nash congruences (3,0) (7,1) (5,1) (73,3) (13,11) (19,15) and (37,27), giving flip congruences (3,1) (7,0) (5,0) (73,7) (13,2) (19,4) and (37,10). Evaluating the flip Sierpinski number, we find k' = 29024869, which has Keller label 9.29. In other words, the flip does not directly match up the Keller labels and requires 29 Keller iterations to do so. We shall call the number of Keller iterations between the base values of the two Keller cycles the flip offset. The flip of 9.0 is, unsurprisingly, found to be 1.29. Rather than slide left 29 places, we could slide right 7 places, by replacing Nash congruences (pi, ri) with (pi, rI+7). However, there is no simple iteration that performs a slide to the right.

In the above example, the number of Keller iterations required is odd, and so there is no label position at which flips match. However, for k = 8184977, labelled 13.0, and which also has covering set B, the flip k' = 66620329 has Keller label 20.16. Since the offset is even, there is a position at which matching labels are flips of each other. This happens at the 8th iteration, that is, Sierpinski numbers 13.8 and 20.8 are flips of each other.

The only problem with our conjecture is the possibility that the flipping process might map every congruence onto itself, and hence k k' (to within Keller cycles). Suppose the flip is self-mapping, and let p be a member of the covering set such that p has largest exponent ep. The Nash congruence (p,r) flips to (p,r+x) for some offset of x. Therefore, we require m+1- r r- x mod ep, that is, 2r 1- x mod ep. Since there must be other Nash congruences covering all positions from 1 to m, there must be another prime, q say, such that q covers position r+1. By symmetry, the congruence (q,r+1) flips to (q,r+x- 1). Hence m+1- (r+1) r+x- 1 mod eq, that is 2r 1- x mod eq . In general, each prime pi p in the covering set covers position r+yi for some value yi, and we find that 2r 1- x mod ei for all i. Now, using the Keller iteration, we can suppose that r = m, since otherwise we can find another member of the Keller cycle for which this is the case. Thus we have x 1 mod ei for all i. Now, if ei is even, x must be odd, and vice-versa, and hence the ei are either all odd or all even. Also, if there is a prime whose exponent is m, we must have x = 1 since by definition we have x < m.

Suppose the ei are all odd. Then the primes 3 and 5 are immediately ruled out of the covering set. In fact, there are comparatively few primes with odd exponent. The first few are 7, 23, 31, 47, 71, 73, 79, 89, 103, 127, 151, 167, 191, 199, 223, 233, 239. The smallest covering sets that do not include 5 are of modulus 72, there being 7 of these, and they all include 3. The smallest covering set that does not include 3 has modulus 180 and has 16 primes, whose product is approximately 2*1030. All covering sets of modulus 180 include either 3 or 5. There are modulus 360 covering sets that contain neither 3 nor 5, but most of the other primes in these sets have even exponent. A covering set made up entirely of primes with odd exponent would have a huge modulus.

Suppose the ei are all even. Then they must all be powers of 2 since otherwise we could replace primes of order ei with primes of odd exponent fi such that fi divides ei. Consider modulus m = 64. There is only one covering set, consisting of all the primes dividing 264 - 1, and one self-flipping Keller cycle, whose smallest value is offset by 29 when flipped.

Proceeding with the flip idea, we can pair up all the known Keller cycles, and also find new ones that lie outwith the current search range. For instance, set H (modulus 48) is associated with only one Keller cycle for k < 108, starting at k = 11206501. The Nash congruences are (3,1) (7,2) (5,2) (17,0) (257,4) (241,12) and (97,28). The flip congruences are (3,0) (7,2) (5,3) (17,1) (257,13) (241,13) and (97,21). Solving the generated simultaneous congruences gives k' = 8407945943. The Keller cycle produced from this has smallest value k'' = 597875869, which is therefore labelled as the zeroth member of the cycle, and k' is 5 Keller iterations from this. Thus we have found a matching Keller cycle for the one we already knew, with an offset of 5.

For completeness, of the 34 Keller cycles found during the search over k < 108, we have the following flipping pairs, using label numbers and giving a flip offset.

A

2 & 3

offset 0

B

1 & 9

offset 29

 

13 & 20

offset 16

C

5 & 10

offset 27

D

6 & 22

offset 20

 

12 & 17

offset 17

E

7 & 14

offset 19

 

11 & 15

offset 33

F

4 & 25

offset 19

 

19 & 32

offset 13

L

24 & 28

offset 7

The two Keller cycles 8 and 26 associated with set G do not constitute a pair, but are both paired with cycles outwith the current search range. The same is true of the two cycles associated with set K, though the standard calculations lead to a base value k = 115281169, not much beyond the current limits, whose Keller cycle pairs with cycle 23.

Note that using the alternative congruences (pi, m- ri) as the flip congruences instead of

(pi, m+1- ri) does not change the pairing of Keller cycles, but increases the offset by 1 (modulo m).

We now continue the search to 2*108. For covering set B, 2P(B) is reached within this range, and no generated values are listed.

100093157

36

prime

E

14.18

100323289

36

11.

E

14.27

100387913

36

prime

B

20.21

100834471

36

prime

B

1.27

102310339

36

 

D

17.20

102790343

36

prime

B

13.2

102832981

36

 

B

20.10

103812287

36

41.

D

6.23

104697533

36

61.

F

19.11

105114931

36

59.

B

1.35

105885947

72

 

R

35.0

106330741

36

11.

B

9.35

106363697

36

11.31.

B

9.20

106596713

36

101.

D

12.35

107177209

36

prime

E

7.1

107432603

36

113.167.

F

25.34

108628343

72

151.

S

36.0

108923657

36

prime

B

1.24

109093577

36

23.

D

17.35

109168141

36

107.197.

B

13.35

109758563

36

 

E

7.2

110213267

36

127.

B

9.4

110271619

36

31.139.157.163

B

1.9

110676233

36

prime

B

13.30

110825251

36

prime

D

12.12

111151351

36

prime

B

9.7

111267859

36

23.

B

9.25

111608297

36

97.

B

13.12

111618167

36

prime

B

1.12

111792841

36

prime

D

22.35

112787573

36

prime

E

7.30

113271289

36

17.223.

B

20.8

114145729

36

53.

B

9.33

114596513

36

337.

F

4.28

114855079

36

101.

B

13.33

114921271

36

 

E or F

7.3 or 32.34

115281169

72

 

K

37.0

115449353

36

 

F

4.12

115566797

72

 

R

35.44

116138629

36

 

D

6.32

116279749

36

 

E

11.1

117188839

36

 

F

32.12

118656023

36

 

B

9.10

118912069

36

 

E

7.21

119588087

36

 

B

9.28

119863547

48

 

C

10.7

119879899

48

 

C

5.24

120527153

36

 

D

12.27

120979291

36

 

E

7.31

121074511

36

 

B

20.2

122091961

48

 

C

10.26

122311103

36

 

D or F

22.18 or 32.21

122311591

36

 

B

13.15

122390551

36

 

B

1.15

122514181

36

 

E

14.1

122685113

36

 

E

7.14

123100501

36

 

E

11.15

124371917

36

 

B

9.12

124463569

36

 

E

7.25

125208229

36

 

E

15.1

125246687

36

 

E

7.4

125773231

36

 

D

6.18

126351319

36

 

D

22.13

126596461

36

 

E

15.23

127127419

36

 

B

9.17

127963643

36

 

E

11.2

128100173

36

 

B

9.30

129197389

36

 

F

32.32

129764281

36

 

F

32.10

130725391

36

 

B

20.22

130896953

36

 

B

20.29

131044847

36

 

F

32.19

131618507

36

 

B

1.28

133228283

36

 

E or F

7.22 or 32.17

134045869

36

 

B

20.4

135147473

36

 

B

20.25

135229937

36

 

F

25.20

135530251

36

 

B

13.3

135615527

36

 

B

20.11

135792317

36

 

E

11.8

136519969

36

 

B

20.32

136616693

36

 

E

15.6

137021401

36

 

B

13.7

137362727

36

 

E

7.32

137536591

36

 

D

17.6

137847349

36

 

E

7.9

138385817

36

 

B

20.15

138411353

36

 

E

11.26

138836071

36

 

D

17.30

138920423

36

 

D

22.30

138994189

36

 

B

13.17

139051463

36

 

F

25.22

139310029

36

 

B

1.17

139323721

36

 

B

13.23

139630819

36

 

F

19.18

140432507

36

 

E

14.2

140733241

36

 

E

11.31

140774371

36

 

E

7.15

141605147

36

 

E

11.16

143453693

36

 

E

15.14

144043891

36

 

D

12.24

144331283

36

 

E

7.26

144975841

36

 

E

11.11

145820603

36

 

E

15.2

145897519

36

 

E

7.5

146053577

72

 

G

38.0

146880319

48

 

C

5.40

148597067

36

 

E

15.24

150553051

36

 

D

17.26

150558323

36

 

D

22.26

151060223

48

 

C

10.13

151331431

36

 

E

11.3

151570849

36

 

E

15.9

152106751

48

 

C

5.30

152252267

36

 

F

19.33

154337567

36

 

D or F

17.19 or 25.24

155088541

36

 

D

6.22

155223473

72

 

S

39.0

155825641

36

 

E

14.25

156654991

36

 

F

19.20

156701453

72

 

P

40.0

157539121

36

 

E

7.35

158595023

36

 

D

12.11

161416097

36

 

E

7.12

161860711

36

 

E

7.23

163378771

48

 

C

5.14

164337949

36

 

D

22.17

165025171

36

 

F

4.25

165347657

36

 

E

11.24

165928129

36

 

E

11.29

166069013

36

 

D

6.17

166358057

36

 

D

22.12

166988779

36

 

E

11.9

167604349

36

 

F

25.17

168637531

36

 

E

15.7

169073869

60

 

T

41.0

169701229

36

 

E

14.23

170129599

36

 

E

7.33

170337737

72

 

K

37.27

171098843

36

 

E

7.10

171950693

36

 

D

17.5

172081733

36

 

E

11.22

172226851

36

 

E

11.27

172600433

36

 

D

17.29

172642609

36

 

D

22.29

174329011

60

 

U

42.0

175204343

36

 

D

12.23

176269159

36

 

E

14.3

176870627

36

 

E

11.32

176952887

36

 

E

7.16

177065453

36

 

E

14.32

178458923

36

 

D

17.25

178461559

36

 

D

22.25

178614439

36

 

E

11.17

180351181

36

 

D

17.18

181040117

36

 

E

15.28

181339441

48

 

C

5.38

182097361

36

 

E

14.5

182311531

36

 

E

15.15

182384417

48

 

C

10.11

182479909

36

 

D or E

12.10 or 14.15

182646049

48

 

C

5.28

184066711

36

 

E

7.27

184503233

36

 

E

11.34

184832273

36

 

E

7.18

185282537

36

 

E

14.34

185355827

36

 

E

11.12

185619293

36

 

E

15.34

187045351

36

 

E

15.3

187199183

36

 

E

7.6

190784569

36

 

D

12.22

191478481

36

 

E

11.19

192411859

36

 

D

17.24

192413177

36

 

D or F

22.24 or 32.27

192598279

36

 

E

15.25

192778253

36

 

E

14.12

198067007

36

 

E

11.4

198545843

36

 

E

15.10

199388327

36

 

D

17.23

This range provides us with the first Sierpinski numbers of modulus 60. Covering sets and sample Sierpinski numbers with modulus 60 have been obtained using alternative methods previously, but the smallest such numbers had not.

The new covering sets are :

code

modulus

covering set

R

72

{3, 5, 7, 13, 17, 37, 109, 433}

S

72

{3, 5, 7, 13, 17, 19, 73, 433}

T

60

{3, 5, 7, 11, 13, 41, 61, 151, 331}

U

60

{3, 5, 7, 11, 13, 31, 41, 61, 331}

Additional overlaps found including new cycles are :

8664092933 - shared by cycles 23 (K) and 39 (S)

9863014727 - shared by cycles 36 (S) and 37 (K)

9936760217 - shared by cycles 36 (S) and 37 (K)

11968332053 - shared by cycles 8 (G) and 35 (R)

21530959441 - shared by cycles 8 (G) and 35 (R)

24733157753 - shared by cycles 8 (G) and 35 (R)

The overlaps involving sets K and S result from interchangeability of the subsets {3,5,37} and (3,5,73}. Since 73 has smaller exponent than 37, set S is the primary covering set. The G & R overlap is caused by interchanging the sets {5,19} and {5,37}.

If we continue, the limit 2P(E) = 209191710 is reached quickly and no new cycles with covering set E are found. The following list takes us as far as 2P(D) = 412729590.

200019049

36

 

F

25.9

201181193

36

 

E

15.30

202876561

36

 

D

17.22

205410169

36

 

E

14.7

206266849

36

 

E

15.17

206940361

36

 

E

14.17

207055427

36

 

E

14.26

207140783

36

 

F

19.35

208234613

36

 

D

6.1

208884353

36

 

D

6.25

210104431

36

 

D

6.2

210465533

72

 

P

43.0

210586403

72

 

P

40.36

211062227

72

 

V

44.0

211073063

72

 

K

23.21

211403911

36

 

D

6.26

212497043

72

 

W

45.0

213447751

48

 

I

21.28

213578567

48

 

I

46.0

213844067

36

 

D

6.3

214767383

72

 

P

43.48

215481983

36

 

F

25.26

216443027

36

 

D

6.27

218039041

48

 

C

10.30

218649563

36

 

F

4.20

220022057

36

 

D

12.1

221323339

36

 

D

6.4

224129461

72

 

G

26.24

224751679

36

 

F

19.22

225632671

48

 

L

24.14

226521259

36

 

D or J

6.28 or 29.9

228505373

48

 

L

28.17

230009513

36

 

D

17.1

230667589

36

 

D

12.6

232190537

48

 

C

10.23

233679319

36

 

D

12.2

235566677

36

 

F

19.29

236281883

36

 

D or J

6.5 or 29.22

236936209

36

 

D

12.14

240806569

36

 

D

22.1

240936503

72

 

K

47.0

245952859

48

 

C

10.20

246677723

36

 

D

6.29

252919021

36

 

D

6.14

253282909

36

 

F

19.10

253424869

48

 

C

10.44

253654231

36

 

D

17.2

254970383

36

 

D

12.7

258189721

36

 

D

6.34

258232399

36

 

F

4.27

258658819

36

 

F

4.11

260993843

36

 

D

12.3

265532837

36

 

F

32.31

265816283

36

 

F

32.9

266142407

60

 

X

48.0

266198971

36

 

D

6.6

267507623

36

 

D

12.15

268549111

36

 

F

25.19

272308613

36

 

D

6.9

275248343

36

 

D

22.2

275743817

36

 

D

12.29

278770729

48

 

Y

49.0

282777829

36

 

D or J

12.18 or 18.13

282879617

36

 

D

22.20

284736317

36

 

F

25.16

286604449

48

 

L

28.44

286990651

36

 

D

6.30

290215763

48

 

L

24.17

291966617

48

 

C

10.47

292099127

60

 

X

50.0

293678719

48

 

C

10.32

296728129

36

 

D

6.20

297140731

36

 

F

32.26

297988073

60

 

X

51.0

299040481

36

 

D

22.15

299473247

36

 

D

6.15

300943667

36

 

D or F

17.3 or 25.8

301946329

48

 

L

28.40

303043789

36

 

D

17.16

303126409

72

 

S

36.51

303575971

36

 

D

12.8

308587501

72

 

N

30.34

308914459

72

 

Z

52.0

310014647

36

 

D

6.35

313197379

48

 

L

28.20

314416021

72

 

K

37.6

315622891

36

 

D

12.4

318717481

36

 

F

19.28

321185749

36

 

D

6.12

326033147

36

 

D

6.7

327575597

36

 

F

19.9

328650451

36

 

D

12.16

332320309

60

 

X

53.0

333700561

36

 

F

32.30

333716941

36

 

D

17.14

338252431

36

 

D

6.10

338513269

48

 

L

28.12

341371831

48

 

AA

54.0

341385229

36

 

D

17.12

343302301

36

 

D or F

17.10 or 25.15

343401623

48

 

L

24.41

343781569

36

 

D

17.8

344131891

36

 

D

22.3

344703589

36

 

D

22.5

345122839

36

 

D

12.30

346990381

36

 

D

22.7

348667381

36

 

D

12.32

348979489

36

 

D

17.32

349316897

36

 

D

22.32

350284703

48

 

C

10.25

351639391

48

 

L

28.26

354064201

72

 

Z

52.38

356137549

36

 

D

22.9

359190863

36

 

D

12.19

359292259

48

 

AB

55.0

359394439

36

 

D

22.21

360292883

36

 

F

19.27

361019063

72

 

G

8.56

362845549

36

 

D

12.34

364093981

36

 

D

17.34

364721941

36

 

F

19.8

365443613

36

 

D

22.34

365928503

72

 

AC

56.0

367616507

36

 

D

6.31

367784423

36

 

F

32.29

369699767

72

 

R

57.0

369810769

36

 

D

12.26

372585293

36

 

F

25.14

379908443

48

 

C

5.37

380430931

48

 

C

10.10

380561747

48

 

C

5.27

383295113

36

 

F

19.7

384932701

48

 

L

24.46

387091463

36

 

D

6.21

387226789

36

 

F

25.13

388176109

72

 

P

31.39

391716167

36

 

D

22.16

392581699

36

 

D or F

6.16 or 19.6

392726221

36

 

D

22.11

394547537

36

 

F

25.12

395522539

36

 

D

17.4

395847409

36

 

D

17.28

395868497

36

 

D

22.28

398207911

36

 

F

25.11

398258243

48

 

C

10.29

399722783

36

 

D

17.17

400787147

36

 

D

12.9

402513331

36

 

F

4.1

403158377

36

 

F

4.2

404448469

36

 

F

4.3

404939477

36

 

D

12.21

405333991

48

 

C

10.22

405753781

36

 

D

22.23

407028653

36

 

F

4.4

408182963

72

 

G

26.43

410985473

36

 

D

17.21

412189021

36

 

F

4.5

The new covering sets are :

code

modulus

covering set

V

72

{3, 5, 7, 17, 37, 109, 241, 433}

W

72

{3, 5, 7, 13, 19, 109, 241, 433}

X

60

{3, 5, 7, 11, 13, 31, 41, 61, 151}

Y

48

{3, 5, 7, 13, 97, 241, 673}

Z

72

{3, 5, 13, 17, 19, 37, 109, 241}

AA

48

{3, 5, 7, 17, 97, 257, 673}

AB

48

{3, 5, 13, 17, 97, 241, 673}

AC

72

{3, 5, 7, 13, 19, 37, 241, 433}

Note that we have run out of convenient letters to identify covering sets, and it is appropriate to develop a new nomenclature.

Two of the new covering sets do not include the prime 7. In fact, the value of 52.0 was initially overlooked by the algorithm since it divisible by 7, and only located by generating the Keller cycle from 52.38. It is possible that other Sierpinski numbers have been similarly overlooked, but, as mentioned previously, will eventually be discovered. However, it is likely that 52.0 is the smallest Sierpinski number divisible by 7.

Another interesting development is the occurrence of 4 closely grouped values all with covering set X but all in different cycles.

An additional overlap occurs between cycles with covering sets K & R. Many mnore overlaps between members of Keller cycles and generated values exist.

The current allocation of covering sets to Keller cycles, ignoring set J, is as follows:

set

modulus

cycles

status

A

24

2, 3

(complete)

B

36

1, 9, 13, 20

(complete)

C

48

5, 10

 

D

36

6, 12, 17, 22

(complete)

E

36

7, 11, 14, 15

(complete)

F

36

4, 19, 25, 32

 

G

72

8, 26, 38

 

H

48

16

 

I

48

21, 46

 

K

72

23, 33, 37, 47

 

L

48

24, 28

 

M

48

27

 

N

72

30

 

P

72

31, 40, 43

 

R

72

35, 57

 

S

72

36, 39

 

T

60

41

 

U

60

42

 

V

72

44

 

W

72

45

 

X

60

48, 50, 51, 53

 

Y

48

49

 

Z

72

52

 

AA

48

54

 

AB

48

55

 

AC

72

56

 

We now continue to 5*108.

415951157

48

 

C

10.43

422509757

36

 

F

4.6

422590909

72

 

P

31.15

425780911

48

 

I

46.35

426694847

36

 

F

19.1

435119569

48

 

L

24.12

435222031

48

 

C

10.46

435711979

36

 

F

19.14

443151229

36

 

F

4.7

450231953

72

 

G

26.51

451521409

36

 

F

19.2

456324301

48

 

I

46.27

457592539

36

 

F

25.1

460059647

36

 

F

25.28

464561807

48

 

L

24.39

466621249

48

 

L

28.24

469555673

36

 

F

19.15

472729967

36

 

F

4.22

478078081

48

 

AA

58.0

482420221

72

 

G

59.0

484434173

36

 

F

4.8

488303521

60

 

X

60.0

489719779

36

 

F

25.5

494851853

48

 

L

24.37

497138431

36

 

F

19.24

497214301

48

 

C

10.42

497369479

60

 

X

61.0

There are 3340 distinct Sierpinski numbers identified to this point, of which 657 belong to at least one Keller cycle. This includes only one, namely 308914459, that is divisible by 7.

Complete listing of Sierpinski numbers

URL : www.glasgowg43.freeserve.co.uk/siernash.htm