Somer–Lucas pseudoprime

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

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

where ${\displaystyle \left({\frac {D}{N}}\right)}$ 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). "On Lucas d-Pseudoprimes". In Bergum, Gerald E.; Philippou, Andreas N.; Horadam, A. F. (eds.). Applications of Fibonacci Numbers. Vol. 7. Springer Netherlands. pp. 369–375. doi:10.1007/978-94-011-5020-0_41. ISBN 978-94-010-6107-0.
• Carlip, Walter; Somer, Lawrence (2007). "Square-free Lucas d-pseudoprimes and Carmichael-Lucas numbers". Czechoslovak Mathematical Journal. 57 (1): 447–463. doi:10.1007/s10587-007-0072-6. hdl:10338.dmlcz/128183. S2CID 120952494.
• 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. ISBN 9780387944579.