Prime Lattices and the Structure of Arithmetic: A Conceptual Note

This paper gives a clear account of how prime numbers form the basic structure of arithmetic. Using the Fundamental Theorem of Arithmetic, I show that every natural number can be written as a product of primes and that this makes it possible to picture numbers as points in a lattice, each one defined by its prime factors. In this way, arithmetic is not built from isolated numbers but from the network of relations among primes. What is real, on this view, is not the numbers themselves but the structure that connects them.