What question did this study set out to answer?

The aim is to illustrate how prime numbers form the foundational structure of arithmetic through their relationships.

February 26, 2026Open Access

Prime Lattices and the Structure of Arithmetic: A Conceptual Note

Key Points

The aim is to illustrate how prime numbers form the foundational structure of arithmetic through their relationships.
Analyzed the Fundamental Theorem of Arithmetic
Explored the representation of numbers as products of primes
Visualized numbers as points in a lattice
Examined the network of relations among primes
Established that every natural number can be expressed as a product of primes
Demonstrated the interconnectedness of numbers through prime factors
Highlighted that arithmetic arises from the structure connecting primes rather than isolated numbers

Abstract

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.

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Joshua Sanctus

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Prime Lattices and the Structure of Arithmetic: A Conceptual Note

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Abstract

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