Previous |  Up |  Next


Lucas; Fibonacci; stability; uniform distribution; recurrence
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.
[1] R. T. Bumby: A distribution property for linear recurrence of the second order. Proc. Amer. Math. Soc. 50 (1975), 101–106. DOI 10.1090/S0002-9939-1975-0369240-X | MR 0369240 | Zbl 0318.10006
[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
[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
[4] D. H. Lehmer: An extended theory of Lucas’ functions. Ann. of Math. 31 (1930), 419–448. DOI 10.2307/1968235 | MR 1502953
[5] William J. LeVeque: Fundamentals of Number Theory. Dover Publications Inc., Mineola, NY, 1996, Reprint of the 1977 original. MR 1382656
[6] E. Lucas: Théorie des fonctions numériques simplement périodiques. Amer. J. Math. 1 (1878), 184–240, 289–321. DOI 10.2307/2369373 | MR 1505176
[7] L. M. Milne-Thompson: The Calculus of Finite Differences. Macmillan, London, 1951. MR 0043339
[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. MR 0967952 | Zbl 0678.10011
[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
[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
[11] Lawrence Somer: Solution to problem H-377. Fibonacci Quart. 24 (1986), 284–285.
[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
[13] Lawrence Somer, Walter Carlip: Stability of second-order recurrences modulo $p^r$. Int. J. Math. Math. Sci. 23 (2000), 225–241. DOI 10.1155/S0161171200003240 | MR 1757803
[14] Morgan Ward: The arithmetical theory of linear recurring series. Trans. Amer. Math. Soc. 35 (1933), 600–628. DOI 10.1090/S0002-9947-1933-1501705-4 | MR 1501705
[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
Partner of
EuDML logo