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Some Remarks on the Distribution of Second Order Recurrences and a Related Group Structure

In: Applications of Fibonacci Numbers

Author

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  • John R. Burke

Abstract

The distribution properties of the Fibonacci numbers have been closely examined by several authors and from several different viewpoints. One of the earliest results, due to Kuipers and Shiue [5], established that the Fibonacci numbers are uniformly distributed mod p (u.d. mod p) if and only if p = 5. They conjectured that the Fibonacci numbers were u.d. mod for each h ⪰ 1. Niederreiter established the conjecture in [9]. Kuipers and Shiue went on to consider the distribution of general second order recurrences [6]. In 1975, Webb and Long (W-L) characterized all second order recurrences which were u.d. mod ph [13]. (Similar results were established independently by Bumby [1] and Nathanson [8]. See also [2, 7, 10, 11].) In the following a relationship between u.d. mod p second order recurrences (when they exist) and those sequences satisfying the same recurrence relation but are not u.d. mod p will be established. It will be shown that there is a rather simple group structure in which the u.d. mod p sequences and those that are not form the elements. There is also a related result that we will obtain about the independence mod p of certain second order linear recurrences.

Suggested Citation

  • John R. Burke, 1996. "Some Remarks on the Distribution of Second Order Recurrences and a Related Group Structure," Springer Books, in: Gerald E. Bergum & Andreas N. Philippou & Alwyn F. Horadam (ed.), Applications of Fibonacci Numbers, pages 47-52, Springer.
    • Handle: RePEc:spr:sprchp:978-94-009-0223-7_4
    DOI: 10.1007/978-94-009-0223-7_4
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