Anti-Elite Prime Search

Complete to 1E14

Dennis R. Martin

DP Technology Corp., Camarillo, CA

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 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 non-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	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 r_s = 2, the anti-elite prime p divides the Fermat number F_m–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 Křížek, 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 Ф(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://www.research.att.com/~njas/sequences/.

[7] M. Křížek, 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.

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