Note from OEIS Editor: This is a cached copy of Dennis R. Martin's "Elite Prime Search", obtained from Internet Archive with URL http://web.archive.org/web/20131208103921/http://primenace.com/papers/math/ElitePrimes.htm, and stored with permission of the author at OEIS. The name of this document should stay as a102742.html so that the links from other two cached documents will work as expected.


Elite Prime Search

 

Complete to 1E14

 

Dennis R. Martin

DP Technology Corp., Camarillo, CA

dennis.martin@dptechnology.com

 

There are 29 elite primes less than 5E12 but no more between 5E12 and 1E14. As of February 4, 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 M�ller [1] and M�ller [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 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

3

2

1

1

-1

3

2

5

2

2

1

-1

5

3

7

3

0

2

-1

7

4

41

17

2

4

-1

41

5

15361

8937

8

4

-1

1

6

23041

9692

5

4

-1

1

7

26881

2767

6

4

-1

1

8

61441

60426

10

4

-1

1

9

87041

41476

9

8

-1

161

10

163841

74103

14

4

-1

161

11

544001

221652

7

8

-1

161

12

604801

360626

6

6

-1

1

13

6684673

3689982

16

8

-1

193

14

14172161

9839337

13

4

-1

161

15

159318017

127581454

13

8

-1

17

16

446960641

264969553

7

4

-1

1

17

1151139841

861720124

12

4

-1

1

18

3208642561

1375589497

20

4

-1

1

19

38126223361

9882944985

22

4

-1

1

20

108905103361

39278554590

21

4

-1

1

21

171727482881

161773654636

8

8

-1

161

22

318093312001

287792761689

13

4

-1

1

23

443069456129

84762800455

6

8

-1

209

24

912680550401

361573893908

30

4

-1

161

25

1295536619521

1144718793846

25

4

-1

1

26

1825696645121

980255578452

25

4

-1

161

27

2061584302081

967053839472

35

4

-1

1

28

2769999339521

1001509752664

12

4

-1

161

29

3580135407617

123155921421

15

8

-1

17

 

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 must be anti-elite. 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 29 elite primes is 0.7007640115758556998739637117�. This sum has been proven to be convergent by K�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. It is interesting to note that thirteen of the first 29 elite primes have a residue of 1 (mod 240), and therefore a residue of 1 (mod 120), while eight others have a residue of 161 (mod 240), which when combined with 41 gives nine elite primes that have a residue of 41 (mod 120). Since no p > 5  can have a residue of 3 or 5 (mod 30), only four other possible residues are represented in this column, with 17 being the only other residue modulo 240 that is repeated out of the first 29. Out of (240) = 64 residue classes 26 cannot be elite, leaving 38 possible elite residues modulo 240. But of those 38 only 7 have appeared so far.

 

For a list of anti-elite primes, see the corresponding Anti-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; �eber 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: https://oeis.org/.

 

[7] M. K�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, Anti-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.