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February 26, 2026˜The œreasoner0 citationsOpen Access

Prime Lattices and the Structure of Arithmetic: A Conceptual Note

JSJoshua Sanctus

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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Cite This Study

Joshua Sanctus (2026) studied this question.

synapsesocial.com/papers/699fe28895ddcd3a253e64echttps://doi.org/10.54103/1757-0522/29911
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Also Consider

Synapse has enriched 5 closely related papers on similar clinical questions. Consider them for comparative context:

  1. 1The Limits of Arithmetic Language: A Philosophical and Structural Preface to the Prime-Index Isomorphic Arithmetic2026
  2. 2A Unified Structure Theory of Prime Numbers - Conceptual Framework and Fundamental Path2026
  3. 3Geometric Reconstruction of Prime Structure and Unified Analysis of Number-Theoretic Propositions via Discrete Primitive Growth Space2026
  4. 4Primality as Cyclotomic Saturation: Primes as Finite Fields, Composites as Cartesian Products, and the Structural Blindness of Peano Arithmetic2026
  5. 5Primality as Cyclotomic Saturation: Primes as Finite Fields, Composites as Cartesian Products, and the Structural Blindness of Peano Arithmetic2026