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.

References

• Somer, Lawrence (1998). Bergum, Gerald E.; Philippou, Andreas N.; Horadam, A. F. (eds.). "On Lucas d-Pseudoprimes". Applications of Fibonacci Numbers. 7. Springer Netherlands: 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. ed.). New York: Springer-Verlag. pp. 131–132. {{cite book}}: |edition= has extra text (help)