What does this research mean for the field?

Embedding integers into an infinite-dimensional Euclidean space based on their prime exponents reveals that primes systematically minimize local geometric tension relative to neighboring composites. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.ESTABLISHES_NEW_DIRECTION.

What question did this study set out to answer?

The aim is to develop a geometric framework that represents positive integers and their multiplicative structures using vector spaces.

May 29, 2026Open Access

A Geometric Theory of the Integers in Multiplicative Space

Key Points

The aim is to develop a geometric framework that represents positive integers and their multiplicative structures using vector spaces.
Established a framework where integers are represented by prime exponents in a vector form.
Developed concepts such as multiplicative geodesics and tension functions to study local geometry.
Performed extensive computations and statistical analyses to validate the geometric properties of primes.
Primes minimize local tension relative to composites, with significance below 10^{-280}.
Fourier spectrum of the tension sequence shows a peak at frequency 1/2, emphasizing the prime 2's importance.
Identified connections to the Riemann zeta function and additive combinatorics through multiplicative Laplacian.

Abstract

We develop a systematic geometric framework in which every positive integer is represented by the vector of its prime exponents, Φ (n) = (e₁, e₂, e₃, …) where n = ∏ pᵢ^eᵢ. Under this embedding, multiplication becomes vector addition, divisibility becomes coordinate-wise partial order, primes become canonical unit basis vectors, and the full multiplicative structure of the integers acquires a natural infinite-dimensional Euclidean geometry. We establish the foundational theory of this multiplicative geometry: primes are unit vectors; composites have strictly larger norm; primes are geometrically irreducible; multiplicative distance defines a natural arithmetic metric. We develop the theory of multiplicative geodesics, the multiplicative Laplacian, arithmetic curvature, and weighted prime metrics. We introduce shell geometry in multiplicative space and relate it to classical norm-counting functions. The integer sequence 1, 2, 3, … defines a discrete trajectory in multiplicative space whose local geometry is captured by the tension function T (n) = ∥Φ (n) - Φ₁₆ (n) ∥, measuring deviation from the local arithmetic background. Extensive computation and statistical analysis establish that primes systematically minimise local tension relative to neighbouring composites, with Kolmogorov–Smirnov significance below 10^−280. The Fourier spectrum of the tension sequence exhibits a dominant peak at frequency 1/2, reflecting the structural primacy of the prime 2 as a spectral resonance of the integer lattice. We further develop connections with the Riemann zeta function via spectral zeta functions of the multiplicative Laplacian, with additive combinatorics via Freiman-type structure theorems in multiplicative space, and with analytic number theory via logarithmic embeddings. Several open problems of independent mathematical interest are identified. Mathematics Subject Classification (2020): 11A51, 11C20, 11N99, 47A10, 05C12, 11M41

A Geometric Theory of the Integers in Multiplicative Space

Key Points

Abstract

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