Somer–Lucas pseudoprime explained

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(1)/(\left(N - \left( d) (D)/(N) \right)\right), where \left( (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

• Book: Somer, Lawrence . Applications of Fibonacci Numbers . On Lucas d-Pseudoprimes . 1998 . Bergum, Gerald E. . Philippou, Andreas N. . Horadam, A. F. . 7 . Springer Netherlands . 369–375 . 10.1007/978-94-011-5020-0_41. 978-94-010-6107-0 .
• Square-free Lucas d-pseudoprimes and Carmichael-Lucas numbers . Carlip, Walter . Somer, Lawrence . Czechoslovak Mathematical Journal . 57 . 1 . 2007 . 447–463 . 10.1007/s10587-007-0072-6 . 10338.dmlcz/128183 . 120952494 . free .
  • Book: Ribenboim, P. . §2.X.D Somer-Lucas Pseudoprimes . https://books.google.com/books?id=72eg8bFw40kC&pg=PA131 . The New Book of Prime Number Records . 3rd . New York . Springer-Verlag . 131–132 . 1996. 9780387944579 .