| Title:
|
Square-free Lucas $d$-pseudoprimes and Carmichael-Lucas numbers (English) |
| Author:
|
Carlip, W. |
| Author:
|
Somer, L. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
57 |
| Issue:
|
1 |
| Year:
|
2007 |
| Pages:
|
447-463 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Lucas |
| Keyword:
|
Fibonacci |
| Keyword:
|
pseudoprime |
| Keyword:
|
Fermat |
| MSC:
|
11A51 |
| MSC:
|
11B37 |
| MSC:
|
11B39 |
| idZBL:
|
Zbl 1174.11016 |
| idMR:
|
MR2309977 |
| . |
| Date available:
|
2009-09-24T11:46:59Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128183 |
| . |
| Reference:
|
[1] R. Baillie, S. S. Wagstaff, Jr.: Lucas pseudoprimes.Math. Comput. 35 (1980), 1391–1417. MR 0583518, 10.1090/S0025-5718-1980-0583518-6 |
| Reference:
|
[2] J. Brillhart, D. H. Lehmer, and J. L. Selfridge: New primality criteria and factorizations of $2^m\pm 1$.Math. Comput. 29 (1975), 620–647. MR 0384673 |
| Reference:
|
[3] W. Carlip, E. Jacobson, and L. Somer: Pseudoprimes, perfect numbers, and a problem of Lehmer.Fibonacci Quart. 36 (1998), 361–371. MR 1640372 |
| Reference:
|
[4] W. Carlip, L. Somer: Primitive Lucas $d$-pseudoprimes and Carmichael-Lucas numbers.Colloq. Math (to appear). MR 2291618 |
| Reference:
|
[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. MR 1689539 |
| Reference:
|
[6] R. D. Carmichael: On the numerical factors of the arithmetic forms $\alpha ^n\pm \beta ^n$.Ann. of Math. (2) 15 (1913), 30–70. MR 1502458 |
| Reference:
|
[7] É. Lucas: Théorie des fonctions numériques simplement périodiques.Amer. J. Math. 1 (1878), 184–240, 289–321. (French) MR 1505176 |
| Reference:
|
[8] P. Ribenboim: The New Book of Prime Number Records.Springer-Verlag, New York, 1996. Zbl 0856.11001, MR 1377060 |
| Reference:
|
[9] J. Roberts: Lure of the Integers.Mathematical Association of America, Washington, DC, 1992. MR 1189138 |
| Reference:
|
[10] L. Somer: On Lucas $d$-pseudoprimes.In: Applications of Fibonacci Numbers, Vol. 7 (Graz, 1996). Kluwer Academic Publishers, Dordrecht (1998), 369–375. Zbl 0919.11008, MR 1638463 |
| Reference:
|
[11] H. C. Williams: On numbers analogous to the Carmichael numbers.Can. Math. Bull. 20 (1977), 133–143. Zbl 0368.10011, MR 0447099, 10.4153/CMB-1977-025-9 |
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