OFFSET
1,2
COMMENTS
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).
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).
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.).
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.
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.
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)).
This sequence is a list of generalized Lucas-Carmichael numbers for D=9697.
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.
The resulting sequence of generalized Lucas-Carmichael numbers is based on work done by L. Babinkostova, B. Bentz, M. I. Hassan, and H. Kim.
LINKS
L. Babinkostova, B. Bentz, M. Hassan, A. Hernández-Espiet, and H. J. Kim, Anomalous Primes and the Elliptic Korselt Criterion. (poster presentation)
R. Baillie and S. S. Wagstaff, Lucas Pseudoprimes, Mathematics of Computation, Vol. 35, (1980), 1391-1417.
D. M. Gordon and C. Pomerance, The distribution of Lucas and elliptic pseudoprimes, Mathematics of Computation, Vol. 57: 196, 825-838.
D. E. Hinkel, An investigation of Lucas sequences, Master's theses, Boise State University (2007).
J. Smith, Solvability characterizations of Pell like equations, Master's theses, Boise State University (2009).
Sage program that computes the terms of the sequence: Generalized Lucas Pseudoprime Program
EXAMPLE
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.
PROG
(SageMath) # A program in SageMath is given in the links section.
CROSSREFS
KEYWORD
nonn
AUTHOR
André Hernández-Espiet, Aug 06 2017
STATUS
approved