Some representations of unlimited natural numbers

Abstract

Based on the work of K. Hrbacek [12], we prove that every unlimited natural number ω is of the form ω=ω₁⋅ω₂+ω₃⋅ω₄ in at least k different ways (k≥1 is limited), where ω_{i}∈N is unlimited and ω_{i}/ω_{j} is appreciable for 1≤i,j≤4. Other similar representations of unlimited natural numbers are also presented.