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An exploration of Nathanson's $g$-adic representations of integers

Greg Bell, Austin Lawson, C. Neil Pritchard and Dan Yasaki

Vol. 15 (2022), No. 5, 727–738
Abstract

We use Nathanson’s g-adic representation of integers to relate metric properties of Cayley graphs of the integers with respect to various infinite generating sets S to problems in additive number theory. If S consists of all powers of a fixed integer g, we find explicit formulas for the smallest positive integer of a given length. This is related to finding the smallest positive integer expressible as a fixed number of sums and differences of powers of g. We also consider S to be the set of all powers of all primes and bound the diameter of this Cayley graph by relating it to Goldbach’s conjecture.

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Keywords
g-adic representation, Cayley graph
Mathematical Subject Classification 2010
Primary: 11B13, 11P81
Secondary: 20F65
Milestones
Received: 17 January 2019
Revised: 5 January 2022
Accepted: 20 February 2022
Published: 3 March 2023

Communicated by Kenneth S. Berenhaut
Authors
Greg Bell
Department of Mathematics and Statistics
The University of North Carolina at Greensboro
NC
United States
Austin Lawson
Department of Mathematics and Statistics
The University of North Carolina at Greensboro
NC
United States
Knoxville, TN
United States
C. Neil Pritchard
Department of Mathematics and Statistics
The University of North Carolina at Greensboro
NC
United States
Greensboro, NC
United States
Dan Yasaki
Department of Mathematics and Statistics
The University of North Carolina at Greensboro
NC
United States