Lucas pseudoprimes

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.