Fibonacci Pseudoprime

Consider a Lucas Sequence with $P>0$ and $Q=\pm 1$. A Fibonacci pseudoprime is a Composite Number $n$ such that

$$V_n\equiv P\ \left({{\rm mod\ } {n}}\right).$$

There exist no Even Fibonacci pseudoprimes with parameters $P=1$ and $Q=-1$ (Di Porto 1993) or $P=Q=1$ (André-Jeannin 1996). André-Jeannin (1996) also proved that if $(P,Q)\not=(1,-1)$ and $(P,Q)\not=(1,1)$, then there exists at least one Even Fibonacci pseudoprime with parameters $P$ and $Q$.

See also Pseudoprime

References

André-Jeannin, R. ``On the Existence of Even Fibonacci Pseudoprimes with Parameters $P$ and $Q$.'' Fib. Quart. 34, 75-78, 1996.

Di Porto, A. ``Nonexistence of Even Fibonacci Pseudoprimes of the First Kind.'' Fib. Quart. 31, 173-177, 1993.

Ribenboim, P. ``Fibonacci Pseudoprimes.'' §2.X.A in The New Book of Prime Number Records, 3rd ed. New York: Springer-Verlag, pp. 127-129, 1996.