Lucas Pseudoprime

When $P$ and $Q$ are Integers such that $D=P^2-4Q\not=0$, define the Lucas Sequence $\{U_k\}$ by

$$U_k ={a^k-b^k\over a-b}$$

for $k\geq 0$, with $a$ and $b$ the two Roots of $x^2-Px+Q=0$. Then define a Lucas pseudoprime as an Odd Composite number $n$ such that $n\notdiv Q$, the Jacobi Symbol $(D/n)=-1$, and $n\vert U_{n+1}$.

There are no Even Lucas pseudoprimes (Bruckman 1994). The first few Lucas pseudoprimes are 705, 2465, 2737, 3745, ... (Sloane's A005845).

See also Extra Strong Lucas Pseudoprime, Lucas Sequence, Pseudoprime, Strong Lucas Pseudoprime

References

Bruckman, P. S. ``Lucas Pseudoprimes are Odd.'' Fib. Quart. 32, 155-157, 1994.

Ribenboim, P. ``Lucas Pseudoprimes (lpsp($P,Q$)).'' §2.X.B in The New Book of Prime Number Records, 3rd ed. New York: Springer-Verlag, p. 129, 1996.

Sloane, N. J. A. Sequence A005845/M5469 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.