Functiones et Approximatio Commentarii Mathematici

Lucas pseudoprimes

Andrzej Rotkiewicz

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Abstract

Theorem on four types of pseudoprimes with respect to Lucas sequences are proved.

If $n$ is an Euler-Lucas pseudoprime with parameters $P$ and $Q$ and $n$ is an Euler pseudoprime to base $Q, (n, P) = 1$, then $n$ is Lucas pseudoprime of four kinds.

Let $U_n$ be a nondegenerate Lucas sequence with parameters $P$ and $Q = ±1, \varepsilon = ±1$. Then, every arithmetic progression $ax + b$, where $(a,b) = 1$ which contains an odd integer $n_0$ with the Jacobi symbol $\left(\frac{D}{n_0}\right)$ equal to $\varepsilon$, contains infinitely many strong Lucas pseudoprimes $n$ with parameters $P$ and $Q = ±1$ such that $\left(\frac{D}{n} = \varepsilon \right)$ which are at the same time Lucas pseudoprimes of each of the four types.

Note

Dedicated to Włodzimierz Staś on the occasion of his 75th birthday

Article information

Source
Funct. Approx. Comment. Math., Volume 28 (2000), 97-104.

Dates
First available in Project Euclid: 29 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.facm/1538186686

Digital Object Identifier
doi:10.7169/facm/1538186686

Mathematical Reviews number (MathSciNet)
MR1823995

Zentralblatt MATH identifier
1161.11304

Subjects
Primary: 11A07: Congruences; primitive roots; residue systems 11A51: Factorization; primality
Secondary: 11B39: Fibonacci and Lucas numbers and polynomials and generalizations

Keywords
pseudoprime Dickson pseudoprime Lucas pseudoprime Euler pseudoprime Lucas sequence

Citation

Rotkiewicz, Andrzej. Lucas pseudoprimes. Funct. Approx. Comment. Math. 28 (2000), 97--104. doi:10.7169/facm/1538186686. https://projecteuclid.org/euclid.facm/1538186686


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