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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 359.10045

**Autor: ** Erdös, Paul; Newman, D.J.

**Title: ** Bases for sets of integers. (In English)

**Source: ** J. Number Theory 9, 420-425 (1977).

**Review: ** If every member of a set A of positive integers can be expressed in the form b+b' for some b, b' in B, then the set B of non-negative integers is called a basis for A. The problem is to find bounds on the size m of the smallest basis B for a set A of type (n,N), i.e. for a set A with n elements, the largest of which is N. It is easy to show that n^{ ½ } \leq m \leq **max****{**n+1,(4N+1)^{ ½ }**}**. The authors show that, for most sets A of type (n,N), m > **max**\left**{**\frac n{log N},\frac{\sqrt N}2\right**}**, so that, for most sets A, the actual value of m is closer to the upper bound than to the lower bound. A special study is then made of the sequence of squares, A = **{**1^{2},2^{2},...,n^{2}**}**, and it is shown that this sequence is untypical since, for arbitrarily large M, n^{2/3-\epsilon} \leq m \leq n/ log ^{m}n. (There is an unfortunate misprint at this point.) Closing remarks include the observation that the value of m depends critically not just on n and N, but on the arithmetical character of the sequence.

**Reviewer: ** I.Anderson

**Classif.: ** * 11B13 Additive bases

11B83 Special sequences of integers and polynomials

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