Uniform distribution of two-term recurrence sequences

Author:
William Yslas Vélez

Journal:
Trans. Amer. Math. Soc. **301** (1987), 37-45

MSC:
Primary 11B50

MathSciNet review:
879561

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Abstract: Let be rational integers and for define . The sequence is clearly periodic modulo and we say that is uniformly distributed modulo if for every , every residue modulo occurs the same number of times in the sequence of residues , where is the period of modulo . If is uniformly distributed modulo then divides , so we write . Several authors have characterized those for which is uniformly distributed modulo . In fact in this paper we will show that a much stronger property holds when , a prime. Namely, if is uniformly distributed modulo with period , then every residue modulo appears exactly once in the sequence , for every . We also characterize those composite for which this more stringent property holds.

**[1]**Lee Erlebach and William Yslas Vélez,*Equiprobability in the Fibonacci sequence*, Fibonacci Quart.**21**(1983), no. 3, 189–191. MR**718204****[2]**Władysław Narkiewicz,*Uniform distribution of sequences of integers in residue classes*, Lecture Notes in Mathematics, vol. 1087, Springer-Verlag, Berlin, 1984. MR**766563**

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DOI:
http://dx.doi.org/10.1090/S0002-9947-1987-0879561-2

Article copyright:
© Copyright 1987
American Mathematical Society