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\(p\)-adic quotient sets. (English) Zbl 1428.11023

Summary: For \(A \subseteq \mathbb{N}\), the question of when \(R(A) = \{a/a' : a, a' \in A\}\) is dense in the positive real numbers \(\mathbb{R}_+\) has been examined by many authors over the years. In contrast, the \(p\)-adic setting is largely unexplored. We investigate conditions under which \(R(A)\) is dense in the \(p\)-adic numbers. Techniques from elementary, algebraic, and analytic number theory are employed. We also pose many open questions that should be of general interest.

MSC:

11B05 Density, gaps, topology
11A07 Congruences; primitive roots; residue systems
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
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Online Encyclopedia of Integer Sequences:

a(n) = 5^(n-1)*(3^n - 1)/2.