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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
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Abstract |
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We use Nathanson’s -adic representation of integers to relate metric properties of Cayley graphs of the integers with respect to various infinite generating sets to problems in additive number theory. If consists of all powers of a fixed integer , 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 . We also consider 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
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Mathematical Subject Classification 2010
Primary: 11B13, 11P81
Secondary: 20F65
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Milestones
Received: 17 January 2019
Revised: 5 January 2022
Accepted: 20 February 2022
Published: 3 March 2023
Communicated by Kenneth S. Berenhaut
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Authors |
| Greg Bell | |
| Department of Mathematics and
Statistics The University of North Carolina at Greensboro NC United States |
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| 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 |
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