Löschian Primes: Some Insights into Individual Prime Numbers via Their Novel Scientific Visualization

This essay introduces an approach to visualizing individual prime numbers by embedding them within a hexagonal/triangular lattice framework inspired by Löschian integers emerging in urban geography. Unlike traditional number theory visualizations that primarily focus on large point sets or asymptotic distributions of primes, this method, labelled primetomography, reveals their individual internal structure and spatial relationships. The core idea is to represent selected primes as Löschian quadratic forms corresponding to hexagonal norm geometry. This strategy allows additive adjacency (twin primes), multiplicative collapse (squaring), and lattice norm geometry visualizations as compatible projections. Explorations include relationships between Löschian and twin, Gaussian, Mersenne, and Sophie Germain/safe primes, illustrating known algebraic properties through alternative geometric projections. A key finding is that twin primes exhibit a sharp geometric realization divergence between the smaller non-Löschian and larger Löschian p in a pair. The promoted methodology uses statistical software and geographic information systems (GISs) to illustrate scale-dependent lattice patterns, providing an exploratory visual framework with potential utility in cryptographic analytics.