Note from OEIS Editor: This is a cached copy of Dennis R. Martin's "Anti-Elite Prime Search", obtained from Internet Archive with URL http://web.archive.org/web/20131208101408/http://primenace.com/papers/math/Anti-ElitePrimes.htm, and stored with permission of the author at OEIS. Some hyperlinks have been changed to reflect the new locations of the documents. The name of this document should stay as a128852.html so that the links from other two cached documents will work as expected.


Anti-Elite Prime Search

 

Complete to 1E14

 

Dennis R. Martin

DP Technology Corp., Camarillo, CA

dennis.martin@dptechnology.com

 

There are 100 anti-elite primes less than 5E12, two more between 5E12 and 1E13, and eleven between 1E13 and 1E14, for a total of 113 anti-elite primes less than 1E14. As of February 12, 2009, this search is complete up to 1E14.

 

A prime number p is elite if only finitely many Fermat numbers Fm = 2^(2m) + 1 are quadratic residues of p, while p is anti-elite if only finitely many Fermat numbers are quadratic non-residues of p. Both elite and anti-elite primes were searched for simultaneously in this study using a method based on articles by Chaumont and Müller [1] and Müller [2]. A detailed description is given under Elite and Anti-Elite Search Methodology.

 

In the table below the Exponent m is the smallest nonnegative integer such that Fm = 2^(2m)+1 rs (mod p), where s represents the start of the of the Fermat period (s S, with S being the latest possible Fermat period start derived from Aigner [4]). The length of the Fermat period L is the smallest positive integer such that Fm+L Fm rs (mod p), and the Jacobi Symbol (rk | Fn) applies to all n m and to all k s (specifically in the interval from s to s + L – 1, after which the residues repeat). Hence the prime p can only be a quadratic non-residue, with (rk | Fn) = -1, for a finite number of Fermat numbers, at most those with n < m.

 

#

Anti-Elite
Prime, p

First Repeating
Residue, rs

Exponent

m

Fermat
Period

L

Jacobi
Symbol

(rk | Fn)

p

mod

240

1

2

1

0

1

1

2

2

13

4

2

2

1

13

3

17

2

3

1

1

17

4

97

62

4

2

1

97

5

193

109

5

2

1

193

6

241

16

3

2

1

1

7

257

2

4

1

1

17

8

641

2

6

1

1

161

9

673

256

4

2

1

193

10

769

361

7

2

1

49

11

2689

2382

5

3

1

49

12

5953

2810

5

5

1

193

13

8929

3034

4

5

1

49

14

12289

6049

11

2

1

49

15

40961

5115

11

4

1

161

16

49921

47279

7

4

1

1

17

61681

257

3

4

1

1

18

65537

2

5

1

1

17

19

101377

99619

9

6

1

97

20

114689

2

13

1

1

209

21

274177

2

7

1

1

97

22

286721

249403

7

4

1

161

23

319489

2

12

1

1

49

24

414721

116621

5

4

1

1

25

417793

321693

12

8

1

193

26

550801

17

2

8

1

1

27

786433

393985

17

2

1

193

28

974849

2

12

1

1

209

29

1130641

257

3

12

1

1

30

1376257

1285807

9

6

1

97

31

1489153

1190947

7

3

1

193

32

1810433

1613557

12

8

1

113

33

2424833

2

10

1

1

113

34

3602561

3296917

6

4

1

161

35

6700417

2

6

1

1

97

36

6942721

1973594

10

4

1

1

37

7340033

5435959

19

3

1

113

38

11304961

7745119

14

4

1

1

39

12380161

2544570

10

4

1

1

40

13631489

2

19

1

1

209

41

15790321

257

3

3

1

1

42

17047297

11264373

7

6

1

97

43

22253377

65536

5

2

1

97

44

26017793

2

13

1

1

113

45

39714817

9977849

14

2

1

97

46

45592577

2

11

1

1

17

47

63766529

2

13

1

1

209

48

67411969

26037099

12

12

1

49

49

89210881

54761830

12

6

1

1

50

93585409

32315515

17

6

1

49

51

113246209

35214048

20

6

1

49

52

119782433

65537

4

10

1

113

53

152371201

82851301

14

2

1

1

54

167772161

2

24

1

1

161

55

171048961

60067527

16

6

1

1

56

185602561

27136218

8

12

1

1

57

377487361

182575938

20

4

1

1

58

394783681

65537

4

4

1

1

59

597688321

242516737

20

2

1

1

60

618289153

351505261

10

12

1

193

61

663239809

208987074

6

6

1

49

62

825753601

2

17

1

1

1

63

902430721

366441810

15

4

1

1

64

1107296257

574691068

24

2

1

97

65

1214251009

2

16

1

1

49

66

2281701377

531923830

25

8

1

17

67

3221225473

1610563585

28

2

1

193

68

4278255361

65537

4

4

1

1

69

4562284561

257

3

4

1

1

70

5733744641

4554956764

14

4

1

161

71

6487031809

2

11

1

1

49

72

6511656961

1113753254

20

4

1

1

73

7348420609

4192688106

22

2

1

49

74

11560943617

2754162610

16

2

1

97

75

15600713729

13409578292

20

14

1

209

76

23447531521

21128413018

9

8

1

1

77

29796335617

22207075727

27

2

1

97

78

30450647041

29227412725

19

10

1

1

79

46908728641

65537

4

4

1

1

80

48919385089

21451561671

8

3

1

49

81

70525124609

2

20

1

1

209

82

74490839041

3563083128

26

2

1

1

83

77309411329

10049501084

29

2

1

49

84

83751862273

77436865084

29

12

1

193

85

96645260801

62551430004

7

4

1

161

86

107767726081

26457611963

15

3

1

1

87

137603804161

118753253404

6

4

1

1

88

190274191361

2

13

1

1

161

89

206158430209

103078821889

35

2

1

49

90

246423748609

200858196534

26

2

1

49

91

448203325441

444809836474

23

2

1

1

92

529566400513

32887465995

16

6

1

193

93

646730219521

2

20

1

1

1

94

1084521185281

680019429390

21

6

1

1

95

1753340313601

27210396312

16

4

1

1

96

2115221118977

773629242648

11

3

1

17

97

2422022479873

2175588931933

12

2

1

193

98

2710954639361

2

14

1

1

161

99

2748779069441

2

37

1

1

161

100

4485296422913

2

22

1

1

113

101

5469640851457

4236441622875

30

2

1

97

102

6597069766657

2

39

1

1

97

103

17317308137473

14222889442147

37

6

1

193

104

25409026523137

2

33

1

1

97

105

25991531462657

2

26

1

1

17

106

28114855919617

24856331238165

31

2

1

97

107

31065037602817

2

18

1

1

97

108

32796705816577

16301685102646

24

6

1

97

109

44479210368001

4294967297

5

4

1

1

110

46179488366593

2

40

1

1

193

111

67280421310721

2

7

1

1

161

112

76861124116481

2

27

1

1

161

113

84885296460737

4294967297

5

18

1

17

?

151413703311361

2

28

1

1

1

?

640126220763137

2

31

1

1

17

?

1095981164658689

2

31

1

1

209

?

1238926361552897

2

9

1

1

17

?

1256132134125569

2

13

1

1

209

?

2327042503868417

2

16

1

1

17

?

2405286912458753

2

30

1

1

113

?

2917004348489729

2

39

1

1

209

 

Note that the first two Fermat primes, F0 = 3 and F1 = 5, are elite, while all other Fermat primes as well as all prime divisors of composite Fermat numbers are anti-elite. When rs = 2, the anti-elite prime p divides the Fermat number Fm–1. Out of the first 100 anti-elite primes, 24 of them are either Fermat primes or divisors of composite Fermat numbers and 76 are not, while of the first 113 anti-elite primes there are 31 of them that are Fermat primes or Fermat divisors and 82 that are neither. Prime divisors of Fermat numbers are compiled by Keller [5].

 

The elite and anti-elite primes appear as sequences A102742 and A128852 in Sloane’s Online Encyclopedia of Integer Sequences (OEIS) [6].

 

The sum of the reciprocals of the first 113 anti-elite primes is 0.6644754741377456237324059150…. This value was proven to be convergent in [2] by Chaumont and Müller. The corresponding sum of reciprocals for elite primes has been proven to be convergent by Kķ˛ek, Luca, and Somer [7].

 

The last column in the table gives the residue of p mod 240. These residue classes are used as part of the Elite and Anti-Elite Search Methodology. Out of (240) = 64 residue classes 16 cannot be anti-elite, leaving 48 possible anti-elite residues modulo 240. It is interesting to note that of those 48 only 9 have appeared so far: {1, 13, 17, 49, 97, 113, 161, 193, 209}, and 13 has only appeared for 13 itself.

 

For a list of elite primes, see the corresponding Elite Prime Search page [8].

 

 

References

 

[1] Alain Chaumont and Tom Mueller, All Elite Primes Up to 250 Billion, J. Integer Sequences, Vol. 9 (2006), Article 06.3.8.

 

[2] Tom Mueller, On Anti-Elite Prime Numbers, J. Integer Sequences, Vol. 10 (2007), Article 07.9.4.

 

[3] Chris Caldwell, The Prime Pages: Jacobi symbol.

 

[4] Alexander Aigner; Üeber Primzahlen, nach denen (fast) alle Fermatzahlen quadratische Nichtreste sind. Monatsh. Math. 101 (1986), pp. 85-93.

 

[5] Wilfrid Keller, Fermat factoring status.

 

[6] N. J. A. Sloane, Online Encyclopedia of Integer Sequences (OEIS), electronically published at: http://oeis.org.

 

[7] M. Kķ˛ek, F. Luca, L. Somer, On the convergence of series of reciprocals of primes related to the Fermat numbers. J. Number Theory 97 (2002), 95–112.

 

[8] Dennis R. Martin, Elite Prime Search.

 

 

Copyright © 2008-2009 by Dennis R. Martin, ALL RIGHTS RESERVED.

 

No part of this document may be reproduced, retransmitted, or redistributed by any means, without providing a proper reference crediting Dennis R. Martin.