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.
Dedication
Dedicated to Włodzimierz Staś on the occasion of his 75th birthday
Citation
Andrzej Rotkiewicz. "Lucas pseudoprimes." Funct. Approx. Comment. Math. 28 97 - 104, 2000. https://doi.org/10.7169/facm/1538186686
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