Abstract
TextA set A of nonnegative integers is said to be complete if every sufficiently large natural number is the sum of distinct terms taken from A. In 1959, Birch confirmed a conjecture of Erdős by proving that the set {pnqm:n,m=0,1,…} is complete, where p and q are two coprime integers greater than 1. In this paper, we study extensively the set P(Sp1,…,pr) of all vectors (m1,…,mr) with integer coordinates which can be represented as ∑(p1α1,…,prαr), where p1,…,pr are integers greater than 1 and α1,…,αr are nonnegative integers. We find many regular parts in P(Sp1,…,pr), for example, discrete rectangles. To study the problem, we find a new matching principle. VideoFor a video summary of this paper, please visit https://youtu.be/qXUe1E4jlLQ.
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