Lucas Pseudoprime

When  and  are integers such that , define the Lucas sequence  by

for , with  and  the two roots of . Then define a Lucas pseudoprime as an odd composite number  such that , the Jacobi symbol , and .

The congruence  holds for every prime number , where  is a Lucas number. However, some composites also satisfy this congruence. The Lucas pseudoprimes corresponding to the special case of the Lucas numbers  are those composite numbers  such that . The first few of these are 705, 2465, 2737, 3745, 4181, 5777, 6721, ... (OEIS A005845).

The Wolfram Language implements the multiple Rabin-Miller test in bases 2 and 3 combined with a Lucas pseudoprime test as the primality test in the function PrimeQ[n].

REFERENCES:

Baillie, R. and Wagstaff, S. S. Jr. "Lucas Pseudoprimes." Math. Comput. 35, 1391-1417, 1980.

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

Ribenboim, P. "Lucas Pseudoprimes (lpsp())." §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 "The On-Line Encyclopedia of Integer Sequences."