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

Walter Carlip; Lawrence Somer

Czechoslovak Mathematical Journal (2007)

  • Volume: 57, Issue: 1, page 447-463
  • ISSN: 0011-4642

Abstract

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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.

How to cite

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Carlip, Walter, and Somer, Lawrence. "Square-free Lucas $d$-pseudoprimes and Carmichael-Lucas numbers." Czechoslovak Mathematical Journal 57.1 (2007): 447-463. <http://eudml.org/doc/31141>.

@article{Carlip2007,
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 $(N - \varepsilon (N))/d$, where $\varepsilon $ 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.},
author = {Carlip, Walter, Somer, Lawrence},
journal = {Czechoslovak Mathematical Journal},
keywords = {Lucas; Fibonacci; pseudoprime; Fermat; Fibonacci; pseudoprime},
language = {eng},
number = {1},
pages = {447-463},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Square-free Lucas $d$-pseudoprimes and Carmichael-Lucas numbers},
url = {http://eudml.org/doc/31141},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Carlip, Walter
AU - Somer, Lawrence
TI - Square-free Lucas $d$-pseudoprimes and Carmichael-Lucas numbers
JO - Czechoslovak Mathematical Journal
PY - 2007
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 57
IS - 1
SP - 447
EP - 463
AB - 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 $(N - \varepsilon (N))/d$, where $\varepsilon $ 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.
LA - eng
KW - Lucas; Fibonacci; pseudoprime; Fermat; Fibonacci; pseudoprime
UR - http://eudml.org/doc/31141
ER -

References

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  9. Lure of the Integers, Mathematical Association of America, Washington, DC, 1992. (1992) MR1189138
  10. On Lucas d -pseudoprimes, In: Applications of Fibonacci Numbers, Vol.  7 (Graz, 1996). Kluwer Academic Publishers, Dordrecht (1998), 369–375. (1998) Zbl0919.11008MR1638463
  11. 10.4153/CMB-1977-025-9, Can. Math. Bull. 20 (1977), 133–143. (1977) Zbl0368.10011MR0447099DOI10.4153/CMB-1977-025-9

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