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**Zentralblatt MATH**

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

**Zbl.No: ** 863.11014

**Autor: ** Burr, Stefan A.; Erdös, Paul; Graham, Ronald L.; Li, W.Wen-Ching

**Title: ** Complete sequences of sets of integer powers. (In English)

**Source: ** Acta Arith. 77, No.2, 133-138 (1996).

**Review: ** For a sequence S = (s_{1}, s_{2}, ...) of positive integers, define \Sigma (S): = **{****sum**^{oo}_{i = 1} \epsilon_{i} s_{i}: \epsilon_{i} = 0 or 1, **sum**^{oo}_{i = 1} \epsilon_{i} < oo**}**. Call S complete if \Sigma (S) contains all sufficiently large integers. It has been known for some time that if gcd (a,b) = 1 then the (nondecreasing) sequence formed from the values a^{s} b^{t} with s_{0} \leq s, t_{0} \leq t \leq f (s_{0}, t_{0}) is complete, where s_{0} and t_{0} are arbitrary, and f(s_{0}, t_{0}) is sufficiently large.

In this note we consider the analogous question for sequences formed from pure powers of integers.

**Reviewer: ** S.A.Burr (New York)

**Classif.: ** * 11B83 Special sequences of integers and polynomials

11B13 Additive bases

**Keywords: ** complete sequences; sets of integer powers; bases; sumsets

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