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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A290560 Generalized Lucas-Carmichael numbers for D=9697.

Original entry on oeis.org

1, 35, 143, 323, 385, 455, 595, 665, 899, 935, 1045, 1295, 1547, 1729, 2639, 2737, 2821, 2915, 3289, 3689, 4355, 4465, 5005, 5183, 5291, 6479, 6721, 8855, 8911, 9215, 9361, 10153, 10439, 10465, 11305, 11663, 11951, 15841, 17119, 18095, 19981, 20909, 22607
Offset: 1

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Author

André Hernández-Espiet, Aug 06 2017

Keywords

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.

Examples

			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.

Links

Crossrefs

Cf. A005845, A164824, A217120, A217255, A217719, A227905.

Programs

  • SageMath
    # A program in SageMath is given in the links section.
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