| Title:
|
Bounds for frequencies of residues of second-order recurrences modulo $p^r$ (English) |
| Author:
|
Carlip, Walter |
| Author:
|
Somer, Lawrence |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
132 |
| Issue:
|
2 |
| Year:
|
2007 |
| Pages:
|
137-175 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The authors examine the frequency distribution of second-order recurrence sequences that are not $p$-regular, for an odd prime $p$, and apply their results to compute bounds for the frequencies of $p$-singular elements of $p$-regular second-order recurrences modulo powers of the prime $p$. The authors’ results have application to the $p$-stability of second-order recurrence sequences. (English) |
| Keyword:
|
Lucas |
| Keyword:
|
Fibonacci |
| Keyword:
|
stability |
| Keyword:
|
uniform distribution |
| Keyword:
|
recurrence |
| MSC:
|
11A25 |
| MSC:
|
11A51 |
| MSC:
|
11B37 |
| MSC:
|
11B39 |
| MSC:
|
11B50 |
| idZBL:
|
Zbl 1174.11014 |
| idMR:
|
MR2338803 |
| DOI:
|
10.21136/MB.2007.134189 |
| . |
| Date available:
|
2009-09-24T22:30:25Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134189 |
| . |
| Reference:
|
[1] R. T. Bumby: A distribution property for linear recurrence of the second order.Proc. Amer. Math. Soc. 50 (1975), 101–106. Zbl 0318.10006, MR 0369240, 10.1090/S0002-9939-1975-0369240-X |
| Reference:
|
[2] Walter Carlip, Lawrence Somer: Bounds for frequencies of residues of regular second-order recurrences modulo $p^r$.Number Theory in Progress, Vol. 2 (Zakopane-Kościelisko, 1997), de Gruyter, Berlin, 1999, pp. 691–719. MR 1689539 |
| Reference:
|
[3] R. D. Carmichael: On the numerical factors of the arithmetic forms $\alpha ^n\pm \beta ^n$.Ann. Math. 15 (1913/14), 30–70. MR 1502458 |
| Reference:
|
[4] D. H. Lehmer: An extended theory of Lucas’ functions.Ann. of Math. 31 (1930), 419–448. MR 1502953, 10.2307/1968235 |
| Reference:
|
[5] William J. LeVeque: Fundamentals of Number Theory.Dover Publications Inc., Mineola, NY, 1996, Reprint of the 1977 original. MR 1382656 |
| Reference:
|
[6] E. Lucas: Théorie des fonctions numériques simplement périodiques.Amer. J. Math. 1 (1878), 184–240, 289–321. MR 1505176, 10.2307/2369373 |
| Reference:
|
[7] L. M. Milne-Thompson: The Calculus of Finite Differences.Macmillan, London, 1951. MR 0043339 |
| Reference:
|
[8] H. Niederreiter: A simple and general approach to the decimation of feedback shift-register sequences.Problems Control Inform. Theory/Problemy Upravlen. Teor. Inform. 17 (1988), 327–331. Zbl 0678.10011, MR 0967952 |
| Reference:
|
[9] H. Niederreiter, A. Schinzel, L. Somer: Maximal frequencies of elements in second-order linear recurring sequences over a finite field.Elem. Math. 46 (1991), 139–143. MR 1119645 |
| Reference:
|
[10] Ivan Niven, Herbert S. Zuckerman, Hugh L. Montgomery: An Introduction to the Theory of Numbers, fifth ed.John Wiley & Sons Inc., New York, 1991. MR 1083765 |
| Reference:
|
[11] Lawrence Somer: Solution to problem H-377.Fibonacci Quart. 24 (1986), 284–285. |
| Reference:
|
[12] Lawrence Somer: Upper Bounds for Frequencies of Elements in Second-Order Recurrences Over a Finite Field.Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), Kluwer Acad. Publ., Dordrecht, 1993, pp. 527–546. MR 1271393 |
| Reference:
|
[13] Lawrence Somer, Walter Carlip: Stability of second-order recurrences modulo $p^r$.Int. J. Math. Math. Sci. 23 (2000), 225–241. MR 1757803, 10.1155/S0161171200003240 |
| Reference:
|
[14] Morgan Ward: The arithmetical theory of linear recurring series.Trans. Amer. Math. Soc. 35 (1933), 600–628. MR 1501705, 10.1090/S0002-9947-1933-1501705-4 |
| Reference:
|
[15] William A. Webb, Calvin T. Long: Distribution modulo $p^{h}$ of the general linear second order recurrence.Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 58 (1975), 92–100. MR 0419375 |
| . |
