### Abstract

Let (FORMULA PRESENTED), A, B be rational integers and for (FORMULA PRESENTED). The sequence (un) is clearly periodic modulo m and we say that (un) is uniformly distributed modulo m if for every 5, every residue modulo m occurs the same number of times in the sequence of residues (FORMULA PRESENTED), where N is the period of (u_{n}) modulo m. If (u_{n}) is uniformly distributed modulo m then m divides N, so we write N=mf. Several authors have characterized those m for which (u_{n}) is uniformly distributed modulo m. In fact in this paper we will show that a much stronger property holds when m = p^{k}, p a prime. Namely, if (u_{n}) is uniformly distributed modulo p^{k} with period p^{k}f, then every residue modulo p^{k} appears exactly once in the sequence (FORMULA PRESENTED), for every s. We also characterize those composite m for which this more stringent property holds.

Original language | English (US) |
---|---|

Pages (from-to) | 37-45 |

Number of pages | 9 |

Journal | Transactions of the American Mathematical Society |

Volume | 301 |

Issue number | 1 |

DOIs | |

State | Published - 1987 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**Uniform distribution of two-term recurrence sequences.** / Velez, William Yslas.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 301, no. 1, pp. 37-45. https://doi.org/10.1090/S0002-9947-1987-0879561-2

}

TY - JOUR

T1 - Uniform distribution of two-term recurrence sequences

AU - Velez, William Yslas

PY - 1987

Y1 - 1987

N2 - Let (FORMULA PRESENTED), A, B be rational integers and for (FORMULA PRESENTED). The sequence (un) is clearly periodic modulo m and we say that (un) is uniformly distributed modulo m if for every 5, every residue modulo m occurs the same number of times in the sequence of residues (FORMULA PRESENTED), where N is the period of (un) modulo m. If (un) is uniformly distributed modulo m then m divides N, so we write N=mf. Several authors have characterized those m for which (un) is uniformly distributed modulo m. In fact in this paper we will show that a much stronger property holds when m = pk, p a prime. Namely, if (un) is uniformly distributed modulo pk with period pkf, then every residue modulo pk appears exactly once in the sequence (FORMULA PRESENTED), for every s. We also characterize those composite m for which this more stringent property holds.

AB - Let (FORMULA PRESENTED), A, B be rational integers and for (FORMULA PRESENTED). The sequence (un) is clearly periodic modulo m and we say that (un) is uniformly distributed modulo m if for every 5, every residue modulo m occurs the same number of times in the sequence of residues (FORMULA PRESENTED), where N is the period of (un) modulo m. If (un) is uniformly distributed modulo m then m divides N, so we write N=mf. Several authors have characterized those m for which (un) is uniformly distributed modulo m. In fact in this paper we will show that a much stronger property holds when m = pk, p a prime. Namely, if (un) is uniformly distributed modulo pk with period pkf, then every residue modulo pk appears exactly once in the sequence (FORMULA PRESENTED), for every s. We also characterize those composite m for which this more stringent property holds.

UR - http://www.scopus.com/inward/record.url?scp=84967713025&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84967713025&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-1987-0879561-2

DO - 10.1090/S0002-9947-1987-0879561-2

M3 - Article

AN - SCOPUS:84967713025

VL - 301

SP - 37

EP - 45

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 1

ER -