%I #19 Jun 07 2024 10:50:18
%S 1,35,143,323,385,455,595,665,899,935,1045,1295,1547,1729,2639,2737,
%T 2821,2915,3289,3689,4355,4465,5005,5183,5291,6479,6721,8855,8911,
%U 9215,9361,10153,10439,10465,11305,11663,11951,15841,17119,18095,19981,20909,22607
%N Generalized Lucas-Carmichael numbers for D=9697.
%C On the set Lc(Z/NZ,D) = {(x,y) in (Z/NZ)^2 : x^2 - Dy^2 = 1 (mod N)}, define an operation as follows: (x,y)x(z,w) = (xz+Dyw, xw+zy) (mod N). The set Lc(Z/NZ, D) endowed with this operation is a group. Moreover, the set of Lucas numbers endowed with this operation is a subgroup of Lc(Z/NZ, D).
%C The following results appear in Babinkostova, et al.: If q is a prime, then #Lc(Z/(q^e)Z, D) = (q-(D|q))q^(e-1).
%C The group Lc(Z/(q^e)Z, D) is cyclic for e > 0. This result was proven in Hinkel, 2007 for the case when e = 1. We showed that the statement is true for e > 1 (Babinkostova, et al.).
%C The following notions are introduced in Babinkostova, et al.: A composite integer N is a generalized Lucas pseudoprime (or Lucas pseudoprime in Babinkostova, et al.) to base P in Lc(Z/NZ, D) and integer D if (N-(D|N))P = O, where O is the identity of the group.
%C We define a composite integer N to be a generalized Lucas-Carmichael number if for all P in Lc(Z/NZ, D) it is true that (N-(D|N))P = O.
%C The following Korselt-like criterion holds for a generalized Lucas-Carmichael number: A composite number N is a generalized Lucas-Carmichael number if and only if N is squarefree and for every prime factor q of N, (q-(D|q)) divides (N-(D|N)).
%C This sequence is a list of generalized Lucas-Carmichael numbers for D=9697.
%C For prime values of D less than 10000 and odd nonprime values of N less than 1000000, this is the longest sequence of generalized Lucas-Carmichael numbers.
%C The resulting sequence of generalized Lucas-Carmichael numbers is based on work done by L. Babinkostova, B. Bentz, M. I. Hassan, and H. Kim.
%H L. Babinkostova, B. Bentz, M. Hassan, A. Hernández-Espiet, and H. J. Kim, <a href="http://math.boisestate.edu/reu/publications/KorseltExtension.pdf">Anomalous Primes and the Elliptic Korselt Criterion</a>. (poster presentation)
%H R. Baillie and S. S. Wagstaff, <a href="https://doi.org/10.1090/S0025-5718-1980-0583518-6">Lucas Pseudoprimes</a>, Mathematics of Computation, Vol. 35, (1980), 1391-1417.
%H D. M. Gordon and C. Pomerance, <a href="https://doi.org/10.1090/S0025-5718-1991-1094951-8">The distribution of Lucas and elliptic pseudoprimes</a>, Mathematics of Computation, Vol. 57: 196, 825-838.
%H D. E. Hinkel, <a href="http://scholarworks.boisestate.edu/td/567/">An investigation of Lucas sequences</a>, Master's theses, Boise State University (2007).
%H J. Smith, <a href="http://scholarworks.boisestate.edu/td/55/">Solvability characterizations of Pell like equations</a>, Master's theses, Boise State University (2009).
%H Sage program that computes the terms of the sequence: <a href="https://cocalc.com/projects/fc3a1712-5315-4e9b-979f-758952178f11/files/Sage/OEIS%20code%20Lucas%20groups.sagews">Generalized Lucas Pseudoprime Program</a>
%e We will illustrate an example using the Korselt criterion for generalized Lucas pseudoprimes. Let us observe the second term, 35. Note that 35 = 5*7, so that it is squarefree. Now note that (5-(9697|5)) = 6 and (7-(9697|7)) = 6, both of which divide (35-(9697|35)) = 36. Therefore, by the Korselt criterion for generalized Lucas pseudoprimes, we have that 35 is a generalized Lucas Carmichael number for D = 9697.
%o (SageMath) # A program in SageMath is given in the links section.
%Y Cf. A005845, A164824, A217120, A217255, A217719, A227905.
%K nonn
%O 1,2
%A _André Hernández-Espiet_, Aug 06 2017