Euler-Lucas Pseudoprime

Let $U(P,Q)$ and $V(P,Q)$ be Lucas Sequences generated by $P$ and $Q$, and define

$$D\equiv P^2-4Q.$$

Then

$$\cases{ U_{(n-(D/n))/2}\equiv 0\ \left({{\rm mod\ } {n}}\ri... ...equiv D\ \left({{\rm mod\ } {n}}\right) & when $(Q/n)=-1$,\cr}$$

where $(Q/n)$ is the Legendre Symbol. An Odd Composite Number $n$ such that $(n,QD)=1$ (i.e., $n$ and $QD$ are Relatively Prime) is called an Euler-Lucas pseudoprime with parameters $(P,Q)$.

See also Pseudoprime, Strong Lucas Pseudoprime

References

Ribenboim, P. ``Euler-Lucas Pseudoprimes (elpsp($P,Q$)) and Strong Lucas Pseudoprimes (slpsp($P,Q$)).'' §2.X.C in The New Book of Prime Number Records. New York: Springer-Verlag, pp. 130-131, 1996.