Counterexamples to a Conjecture of Dombi in Additive Number Theory

Jason P. Bell; Jeffrey Shallit

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arXiv · 2212.12473

Counterexamples to a Conjecture of Dombi in Additive Number Theory

Jason P. Bell·Jeffrey Shallit

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Abstract

We disprove a 2002 conjecture of Dombi from additive number theory. More precisely, we find examples of sets $A \subset \mathbb{N}$ with the property that $\mathbb{N} \setminus A$ is infinite, but the sequence $n \rightarrow |\{ (a,b,c) \, : \, n=a+b+c \text{ and } a,b,c \in A \}|$, counting the number of $3$-compositions using elements of $A$ only, is strictly increasing.

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BibTeX RIS

Jason P. Bell, Jeffrey Shallit. 2022-12-23. Counterexamples to a Conjecture of Dombi in Additive Number Theory. https://arxiv.org/abs/2212.12473

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