Elite Prime Search

Complete to 1E14

Dennis R. Martin

DP Technology Corp., Camarillo, CA

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 F_m = 2^(2^m) + 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 F_m = 2^(2^m)+1 r_s (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 F_m+L F_m r_s (mod p), and the Jacobi Symbol (r_k | F_n) 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 (r_k | F_n) = 1, for a finite number of Fermat numbers, at most those with n < m.

#	Anti-Elite Prime, p	First Repeating Residue, r_s	Exponent m	Fermat Period L	Jacobi Symbol *(r_k\| F_n)*	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 Kížek, 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; 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: https://oeis.org/.

[7] M. Kížek, 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.

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