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**Zentralblatt MATH**

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

**Zbl.No: ** 443.10036

**Autor: ** Erdös, Paul; Graham, Ronald L.

**Title: ** On bases with an exact order. (In English)

**Source: ** Acta Arith. 37, 201-207 (1980).

**Review: ** The statement of the results in this paper require the following definitions: Let A be a set of non-negative integers. A is said to be an asymptotic basis of order r (written ord(A) = r) if r is the least integer such that every sufficiently large integer is expressible as the sum of at most r integers from A (allowing repetition). Also, A is said to have exact order s (written ord^*(A) = s) if s is the least integer for which this is possible with exactly s integers from A. There are basis not having exact order. The following results are established:

I. A basis A = **{**a_{1},a_{2},...**}** has an exact order if and only if \gcd**{**a_{k+1}-a_{k}| k = 1,2,...**}** = 1. (1)

II. Let g(r) = **max****{**ord^*(A)**}** subject to (1), and ord(A) = r. Then,

^{1}/_{4} (1+0(1))r^{2} \leq g(r) \leq ^{5}/_{4} (1+0(1))r^{2}.

**Reviewer: ** K.Thanigasalam

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

**Keywords: ** asymptotic basis; exact order

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