Israel Journal of Mathematics

, Volume 57, Issue 2, pp 129–136 | Cite as

Multiplicative representations of integers

  • Melvyn B. Nathanson


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.


Unique Representation Pairwise Disjoint Finite Subset Union Basis Multiplicative System 
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Copyright information

© The Weizmann Science Press of Israel 1987

Authors and Affiliations

  • Melvyn B. Nathanson
    • 1
  1. 1.Office of the Provost and Vice President for Academic AffairsLehman College (CUNY)BronxUSA

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