Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2504.15184

Wave Arithmetic: A Smooth Integral Representation of Number Theory

Stanislav Semenov
View original source ↗Read source document ↗Export metadata ↓

Abstract

We introduce Wave Arithmetic, a smooth analytical framework in which natural, integer, and rational numbers are represented not as discrete entities, but as integrals of smooth, compactly supported or periodic kernel functions. In this formulation, each number arises as the accumulated amplitude of a structured waveform -- an interference pattern encoded by carefully designed kernels. Arithmetic operations such as addition, multiplication, and exponentiation are realized as geometric and tensorial constructions over multidimensional integration domains. Rational numbers emerge through amplitude scaling, and negative values through sign inversion, preserving all classical arithmetic identities within a continuous and differentiable structure. This representation embeds number theory into the realm of smooth analysis, enabling new interpretations of primality, factorization, and divisibility as geometric and spectral phenomena. Beyond technical formulation, Wave Arithmetic proposes a paradigm shift: numbers as the collapsed states of harmonic processes -- analytic resonances rather than atomic symbols.

Explore related subjects

math.GM

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Stanislav Semenov. 2025-04-21. Wave Arithmetic: A Smooth Integral Representation of Number Theory. https://arxiv.org/abs/2504.15184

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Average Chord Lengths in a Triangle

Let $P$ be a point inside a triangle $T$. We consider the average length of the chords of $T$ through $P$, where the direction of the chord is chosen uniformly. An elementary formula is obtained in terms of the distances from $P$ to the sides and vertices of the triangle. Several classical triangle centers give especially simple specializations. For example, if $I$ is the incenter, then \[ M_T(I)=\frac{2r}{\pi} \log\left(\cot\frac A4\cot\frac B4\cot\frac C4\right). \] Our main result is the sharp inequality \[ M_T(P)\le \frac{p}{\pi\sqrt3}\log(2+\sqrt3), \] valid simultaneously for every triangle of perimeter $p$ and every interior point $P$. Thus, among all such pairs $(T,P)$, the largest possible average chord length occurs only when $T$ is equilateral and $P$ is its center. The proof is an elementary symmetrization argument. We close with brief remarks relating the problem to the radial center of a convex body, the electrostatic potential center of a triangle, and dual quermassintegrals.

math.GM↗
arXiv2026

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. We prove a conjecture of Liu stating that \[\sum_{\mathrm{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \geq 2+\left(\frac{2r}{R}\right)^k,~~k>1, \] with the reverse inequality for $0<k<1$. The proof reduces the problem to three positive variables with fixed sum and product. We also determine the equality cases.

math.GM↗