Somer–Lucas pseudoprime

In mathematics, in particular number theory, an odd composite number N is a Somer–Lucas d-pseudoprime (with given d ≥ 1) if there exists a nondegenerate Lucas sequence $U(P,Q)$ with the discriminant $D=P^2-4Q,$ such that $\gcd(N,D)=1$ and the rank appearance of N in the sequence U(P, Q) is

$\frac{1}{d}\left(N-\left(\frac{D}{N}\right)\right),$

where $\left(\frac{D}{N}\right)$ is the Jacobi symbol.

Applications

Unlike the standard Lucas pseudoprimes, there is no known efficient primality test using the Lucas d-pseudoprimes. Hence they are not generally used for computation.

See also

Lawrence Somer, in his 1985 thesis, also defined the Somer d-pseudoprimes. They are described in brief on page 117 of Ribenbaum 1996.

References

• Weisstein, Eric W., "Somer–Lucas Pseudoprime", MathWorld.