Journal of Integer Sequences, Vol. 22 (2019), Article 19.3.7 |

Department of Mathematics

Washington and Lee University

Lexington, VA 24450

USA

**Abstract:**

Given a set *A* of nonnegative integers, define the sum set
*A* + *A* = {*a*_{i} + *a*_{j} | *a*_{i}, *a*_{j} ∈ *A*}
and the difference set
*A* − *A* = {*a*_{i} − *a*_{j} | *a*_{i}, *a*_{j} ∈ *A*}.
The set *A* is said to be sum-dominant if |*A* + *A*| > |*A* − *A*|. In answering a question
by Nathanson, Hegarty used a clever algorithm to find that the smallest cardinality
of a sum-dominant set is 8. Since then, Nathanson has been asking for a
human-understandable proof of the result. We offer a computer-free proof that a set of cardinality less than 6 is not sum-dominant. Furthermore, we prove that the introduction
of at most two numbers into a set of numbers in an arithmetic progression does not
give a sum-dominant set. This theorem eases several of our proofs and may shed light
on future work exploring why a set of cardinality 6 is not sum-dominant. Finally, we
prove that if a set contains a certain number of integers from a specific sequence, then
adding a few arbitrary numbers into the set does not give a sum-dominant set.

Received March 6 2019; revised versions received May 17 2019; May 20 2019.
Published in *Journal of Integer Sequences*,
May 21 2019.

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