Economic hierarchical spatial systems – new properties of Löschian numbers

Löschian numbers are a topic of investigation in scientific fields such as spatial economics, socio-economic geography, geometry, and number theory. This is because these numbers play a significant role in the optimization of spatial interactions, in terms of achieving the best division of space between competing central places. This work presents four new number theoretic theorems and four corollaries concerning the properties of Löschian numbers, which enable a better understanding of the mathematical properties of these numbers. •It has been shown that the two definitions of Löschian numbers given by Konzig and by Dacey are equivalent.•It has been proved that there are essentially only two underlying diophantine equations that generate Löschian numbers.•New theorems and corollaries showing previously unknown properties of Löschian numbers have been formulated and proven.•A formula was established to determine the probability that a randomly chosen natural number is a Löschian number.•The hexagonal networks spatial systems can benefit from the new knowledge concerning Löschian numbers.