Elite and Anti-Elite Prime Search Methodology
Dennis R. Martin
DP Technology Corp.,
dennis.martin@dptechnology.com
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 are being searched for simultaneously in this study using a method based on articles by Chaumont and M�ller [1] and M�ller [2].
Instead of checking whether a
certain congruence is solvable or not for each Fermat residue within the Fermat
period L (as given by expression (4)
in reference [1] or expression (22) in reference [2]), however, the Jacobi symbols of the Fermat residues are
utilized. The algorithm for computing the Jacobi symbol was taken from
An important result from Aigner [4] is that for any prime number written in the form p = 2Sh + 1 with h 1 and odd such that S is a maximum, the S-th term is the latest possible Fermat period start. While the actual period start s could be earlier (s S), the Fermat residue FS must eventually repeat. Aigner also proved in [4] that prime numbers of the form p = 120k + r with k a natural number and r a member of {11, 13, 19, 23, 31, 47, 59, 61, 71, 79, 91, 109, 119} cannot be elite. Chaumont and M�ller in [2] made an analogous proof that prime numbers of the form p = 240k + r with k a natural number and r a member of {7, 23, 43, 47, 67, 83, 103, 107, 127, 143, 163, 167, 187, 203, 223, 227} cannot be anti-elite.
Those results combine to lead to the following algorithm to check the eliteness or anti-eliteness of a candidate prime p:
The results of this currently ongoing search are given on the Elite Prime Search and Anti-Elite Prime Search pages [8, 9]. As of February 12, 2009, this search is complete up to 1E14. There are 29 elite primes less than 1E14 and 113 anti-elite primes less than 1E14.
The elite and anti-elite primes appear as sequences A102742 and A128852 in Sloane�s Online Encyclopedia of Integer Sequences (OEIS) [6].
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, Elite Prime Search.
[9] 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.