Israel Journal of Mathematics

, Volume 57, Issue 2, pp 129–136

# Multiplicative representations of integers

• Melvyn B. Nathanson
Article

## Abstract

Lehh ≧ 2, and let ℬ=(B 1, …,B h ), whereB 1 ⊆ N={1, 2, 3, …} fori=1, …,h. Denote by g(n) the number of representations ofn in the formn=b 1b h , whereb i B i . If v (n) > 0 for alln >n 0, then ℬ is anasymptotic multiplicative system of order h. The setB is anasymptotic multiplicative basis of order h ifn=b 1b n is solvable withb i B for alln >n 0. Denote byg(n) the number of such representations ofn. LetM(h) be the set of all pairs (s, t), wheres=lim g (n) andt=lim g (n) for some multiplicative system ℬ of orderh. It is proved that {fx129-1} In particular, it follows thats ≧ 2 impliest=∞. A corollary is a theorem of Erdös that ifB is a multiplicative basis of orderh ≧ 2, then lim g g(n)=∞. Similar results are obtained for asymptotic union bases of finite subsets of N and for asymptotic least common multiple bases of integers.

## Keywords

Unique Representation Pairwise Disjoint Finite Subset Union Basis Multiplicative System
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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