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Given a finite set $A\subseteq \mathbb{N}$, define the sum set $$A+A =\{a_i+a_j\mid a_i,a_j\in A\}$$ and the difference set $$A-A = \{a_i-a_j\mida_i,a_j\in A\}.$$ The set $A$ is said to be sum-dominant if $|A+A|>|A-A|$.Hegarty used a nontrivial algorithm to find that $8$ is the smallestcardinality of a sum-dominant set. Since then, Nathanson has asked for ahuman-understandable proof of the result. However, due to the complexity of theinteractions among numbers, it is still questionable whether such a proof canbe written down in full without computers' help. In this paper, we present acomputer-free proof that a sum-dominant set must have at least $7$ elements. Wealso answer the question raised by the author of the current paper et al aboutthe smallest sum-dominant set of primes, in terms of its largest element. Usingcomputers, we find that the smallest sum-dominant set of primes has $73$ as itsmaximum, smaller than the value found before.

Sets of Cardinality 6 Are Not Sum-dominant