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Square-free Lucas d-pseudoprimes and Carmichael-Lucas numbers

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Abstract

Let d be a fixed positive integer. A Lucas d-pseudoprime is a Lucas pseudoprime N for which there exists a Lucas sequence U(P, Q) such that the rank of N in U(P, Q) is exactly (N − ε(N))/d, where ε is the signature of U(P, Q). We prove here that all but a finite number of Lucas d-pseudoprimes are square free. We also prove that all but a finite number of Lucas d-pseudoprimes are Carmichael-Lucas numbers.

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References

  1. R. Baillie, S. S. Wagstaff, Jr.: Lucas pseudoprimes. Math. Comput. 35 (1980), 1391–1417.

    Article  MATH  MathSciNet  Google Scholar 

  2. J. Brillhart, D. H. Lehmer, and J. L. Selfridge: New primality criteria and factorizations of 2m ± 1. Math. Comput. 29 (1975), 620–647.

    Article  MATH  MathSciNet  Google Scholar 

  3. W. Carlip, E. Jacobson, and L. Somer: Pseudoprimes, perfect numbers, and a problem of Lehmer. Fibonacci Quart. 36 (1998), 361–371

    MATH  MathSciNet  Google Scholar 

  4. W. Carlip, L. Somer: Primitive Lucas d-pseudoprimes and Carmichael-Lucas numbers. Colloq. Math. To appear.

  5. W. Carlip, L. Somer: Bounds for frequencies of residues of regular second-order recurrences modulo p r. In: Number Theory in Progress, Vol. 2 (Zakopané-Kościelisko, 1997). de Gruyter, Berlin (1999), 691–719.

    Google Scholar 

  6. R. D. Carmichael: On the numerical factors of the arithmetic forms α n ± β n. Ann. of Math. (2) 15 (1913), 30–70.

    Article  MathSciNet  Google Scholar 

  7. É. Lucas: Théorie des fonctions numériques simplement périodiques. Amer. J. Math. 1 (1878), 184–240, 289–321. (In French.)

    Article  MathSciNet  Google Scholar 

  8. P. Ribenboim: The New Book of Prime Number Records. Springer-Verlag, New York, 1996.

    MATH  Google Scholar 

  9. J. Roberts: Lure of the Integers. Mathematical Association of America, Washington, DC, 1992.

    Google Scholar 

  10. L. Somer: On Lucas d-pseudoprimes. In: Applications of Fibonacci Numbers, Vol. 7 (Graz, 1996). Kluwer Academic Publishers, Dordrecht (1998), 369–375.

    Google Scholar 

  11. H. C. Williams: On numbers analogous to the Carmichael numbers. Can. Math. Bull. 20 (1977), 133–143.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Franklin & Marshall College, Lancaster, Pennsylvania, 17604, USA

    W. Carlip

  2. Department of Mathematics, Catholic University of America, Washington, D. C., 20064, USA

    L. Somer

Authors
  1. W. Carlip
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  2. L. Somer
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Carlip, W., Somer, L. Square-free Lucas d-pseudoprimes and Carmichael-Lucas numbers. Czech Math J 57, 447–463 (2007). https://doi.org/10.1007/s10587-007-0072-6

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  • DOI: https://doi.org/10.1007/s10587-007-0072-6

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