A Lucas pseudoprime with parameters 
and 
is an odd composite integer 
satisfying 
and 
, where 
is a positive integer,
is an integer,
, 
is the Jacobi symbol,
and 
denotes the greatest common divisor. The
Lucas sequence 
is given by
 |
where 
and 
and 
are the two polynomial roots of 
. In the commonly used case 
, the defining congruence
becomes 
(Baillie and Wagstaff 1980, Baillie et al. 2021).
Bruckman (1994) uses a different convention. The congruence 
holds for every prime
number 
,
where 
is a Lucas number. The composite
numbers that also satisfy it are called Bruckman-Lucas pseudoprimes. The first
few are 705, 2465, 2737, 3745, 4181, 5777, 6721, ... (OEIS A005845).
This is a test using the Lucas numbers, rather than
the preceding test using the Lucas sequence 
.
The Wolfram Language implements the Miller-Rabin strong pseudoprime tests in bases 2 and 3 combined with a Lucas pseudoprime test as the primality test in the function PrimeQ[n].
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