Counterexample to a Conjecture of Dombi in Additive Number Theory

Jeffrey Shallit

Published 2022-12-23Version 1

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