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

• Somer, Lawrence (1998). Bergum, Gerald E.; Philippou, Andreas N.; Horadam, A. F., eds. "On Lucas d-Pseudoprimes". Applications of Fibonacci Numbers (Springer Netherlands) 7: 369–375. doi:10.1007/978-94-011-5020-0_41.
• Carlip, Walter; Somer, Lawrence (2007). "Square-free Lucas d-pseudoprimes and Carmichael-Lucas numbers". Czechoslovak Mathematical Journal 57 (1).
• Weisstein, Eric W., "Somer–Lucas Pseudoprime", MathWorld.
• Ribenboim, P. (1996). "§2.X.D Somer-Lucas Pseudoprimes". The New Book of Prime Number Records (3rd ed.). New York: Springer-Verlag. pp. 131–132.