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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.