The maximum of the complementary of a semigroup with restricted conditions

TL;DR

This paper investigates the properties of a specific set defined by linear combinations with coprimality and inequality constraints, proving it is unbounded and identifying a particular number beyond which all larger numbers are included.

Contribution

It establishes the unboundedness of the set and identifies a specific number after which all larger integers belong to the set.

Findings

The set  is unbounded.

There exists a natural number M=1674 such that all larger numbers are in .

The set's structure under given conditions is characterized.

Abstract

We consider the set $$\mathcal{A} = \left\{10\cdot a + 11\cdot b \ | \gcd(a,b)=1, a\geq 1, b\geq 2a+1 \right\}.$$ We will prove that $\mathcal{A}$ is unbounded and that there exists a natural number $M\notin \mathcal{A}$ for which $$\left\{M+m:m\geq 1,m\in\mathbb N\right\}\subset \mathcal{A}.$$ Indeed, such number is $M = 1674$.