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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.
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