# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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## Uniform distribution of two-term recurrence sequencesHTML articles powered by AMS MathViewer

by William Yslas Vélez
Trans. Amer. Math. Soc. 301 (1987), 37-45 Request permission

## Abstract:

Let \${u_0}, {u_1}, A, B\$ be rational integers and for \$n \geqslant 2\$ define \${u_n} = A{u_{n - 1}} + B{u_{n - 2}}\$. The sequence \$({u_n})\$ is clearly periodic modulo \$m\$ and we say that \$({u_n})\$ is uniformly distributed modulo \$m\$ if for every \$s\$, every residue modulo \$m\$ occurs the same number of times in the sequence of residues \${u_s}, {u_{s + 1}}, \ldots , {u_{s + N - 1}}\$, 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 \${u_s}, {u_{s + f}}, \ldots , {u_{s + ({p^k} - 1)f}}\$, for every \$s\$. We also characterize those composite \$m\$ for which this more stringent property holds.
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