Counterexamples to a conjecture of Dombi in additive number theory
Tóm tắt
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
$$n \rightarrow |\{ (a,b,c): \ n=a+b+c $$
and
$$ a,b,c \in A \}|$$
counting the number of
$$3$$
-compositions using elements of
$$A$$
only, is strictly
increasing.
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