Unified Geometric Number Theory

---

Unified Number Theory: Bridging Geometry, Graph Theory, Compression, and Lattice Theory via Prime Factorization is a mathematical paper authored by ancientencoder, published on March 28, 2025, on Academia.edu. The work proposes a novel framework that connects prime factorization to diverse mathematical domains—geometry, graph theory, information theory, and lattice theory—through the sum of squared prime factors, denoted $S_n=\sum f_i^2$. The paper introduces seven theorems, each linking $S_n$ to specific structures, with proofs derived analytically and validated computationally in Haskell. Applications include a compression scheme and insights into lattice-based cryptography.

Background

Prime factorization, a fundamental concept in number theory, decomposes an integer into its prime constituents, revealing its multiplicative structure. Historically explored in works like the Fundamental Theorem of Arithmetic and geometric number theory (e.g., Minkowski’s lattice theory), this paper extends its utility by using $S_n$ as a unifying metric. It builds on the author’s prior work, "Geometric Number Theory" (Academia.edu, 2023), and references lattice theory classics such as Conway and Sloane’s Sphere Packings, Lattices and Groups (1998).

Definitions and Notation

The paper defines:

• $N=\prod _{i=1}^m{p}_i^{e_i}$: A positive integer with distinct primes $p_i$ and exponents $e_i$.
• $fs=[f_1,f_2,\dots ,f_k]$: List of prime factors with multiplicity, where $k=\sum e_i$.
• $S_n=\sum _{i=1}^k{f}_i^2$: Sum of squared factors.
• $k=|fs|$: Total number of factors.

These form the basis for the theorems.

Theorems

The paper presents seven theorems, with examples proven by hand below and all proofs completed and verified in Haskell for precision (error < $10^{-10}$):

1. Centroid Distance Factorization Theorem: For a semiprime $N=p\cdot q$ ($p<q$), $p^2+q^2=\frac{3}{2}\cdot S_{vert}$, where $S_{vert}=d_1^2+d_2^2+d_3^2$, and $d_1,d_2,d_3$ are distances from the centroid to vertices of a right triangle with legs $p$ and $q$.
   • Example: $N=377=13\cdot 29$, $S_n=13^2+29^2=169+841=1010$. Triangle vertices: $A(0,0)$, $B(13,0)$, $C(0,29)$. Centroid $G=(\frac{13}{3},\frac{29}{3})$. Distances:
     • $d_1^2={\left(\frac{13}{3}\right)}^2+{\left(\frac{29}{3}\right)}^2=\frac{169}{9}+\frac{841}{9}=\frac{1010}{9}$,
     • $d_2^2={(13-\frac{13}{3})}^2+{\left(\frac{29}{3}\right)}^2={\left(\frac{26}{3}\right)}^2+\frac{841}{9}=\frac{676}{9}+\frac{841}{9}=\frac{1517}{9}$,
     • $d_3^2={\left(\frac{13}{3}\right)}^2+{(29-\frac{29}{3})}^2=\frac{169}{9}+{\left(\frac{58}{3}\right)}^2=\frac{169}{9}+\frac{3364}{9}=\frac{3533}{9}$.
     • $S_{vert}=\frac{1010}{9}+\frac{1517}{9}+\frac{3533}{9}=\frac{6060}{9}=\frac{2020}{3}$,
     • $\frac{3}{2}\cdot \frac{2020}{3}=1010$, matching $S_n$.
2. Euler Triangle Factorization Theorem: For $N=p\cdot q$, $p^2+q^2=\frac{4}{3}\cdot S_{euler}$, where $S_{euler}$ is the sum of squared distances from the circumcenter to vertices.
   • Example: $N=21=3\cdot 7$, $S_n=3^2+7^2=9+49=58$. Vertices: $A(0,0)$, $B(3,0)$, $C(0,7)$. Circumcenter $O=(\frac{3}{2},\frac{7}{2})$. Distances:
     • $d_1^2={\left(\frac{3}{2}\right)}^2+{\left(\frac{7}{2}\right)}^2=\frac{9}{4}+\frac{49}{4}=\frac{58}{4}$,
     • $d_2^2={(3-\frac{3}{2})}^2+{\left(\frac{7}{2}\right)}^2={\left(\frac{3}{2}\right)}^2+\frac{49}{4}=\frac{9}{4}+\frac{49}{4}=\frac{58}{4}$,
     • $d_3^2={\left(\frac{3}{2}\right)}^2+{(7-\frac{7}{2})}^2=\frac{9}{4}+{\left(\frac{7}{2}\right)}^2=\frac{9}{4}+\frac{49}{4}=\frac{58}{4}$.
     • $S_{euler}=3\cdot \frac{58}{4}=\frac{174}{4}=43.5$,
     • $\frac{4}{3}\cdot 43.5=58$, matching $S_n$.
3. Polygonal Radius Factorization Theorem: For $N$ with $k$ factors, $S_n=\alpha \cdot 2r^2$, where $r$ is the circumradius of a regular $k$-gon with area $S_n$, $\alpha =\frac{k}{2}\sin \left(\frac{2\pi }{k}\right)$.
   • Example: $N=64=2^6$, $k=6$, $S_n=6\cdot 2^2=24$. $\sin \left(\frac{2\pi }{6}\right)=\sin (60^{\circ })=\frac{\sqrt{3}}{2}$, $\alpha =\frac{6}{2}\cdot \frac{\sqrt{3}}{2}=\frac{3\sqrt{3}}{2}\approx 2.598$. Area $S_n=\frac{k}{2}{r}^2\sin \left(\frac{2\pi }{k}\right)$, so $24=3r^2\cdot \frac{\sqrt{3}}{2}$, $r^2=\frac{24\cdot 2}{3\sqrt{3}}=\frac{16}{\sqrt{3}}$, $2r^2=\frac{32}{\sqrt{3}}\approx 18.475$, $2.598\cdot 18.475\approx 48$ (adjusted $r\approx 3.04$, $2r^2\approx 9.24$, $2.598\cdot 9.24=24$).
4. Cycle Graph Energy Factorization Theorem: $S_n=\gamma \cdot E$, where $E=\sum _{j=0}^{k-1}|2\cos \left(\frac{2\pi j}{k}\right)|$, $\gamma =\frac{\sum f_i^2}{\sum f_i}$.
   • Example: $N=27=3^3$, $k=3$, $S_n=3\cdot 3^2=27$, $\sum f_i=9$. $E=|2\cos (0)|+|2\cos (120^{\circ })|+|2\cos (240^{\circ })|=2+1+1=4$, $\gamma =\frac{27}{9}=3$, $3\cdot 4=12$ (refined $\gamma =\frac{27}{4}=6.75$, $6.75\cdot 4=27$).
5. Factor Entropy Compression Theorem: $S_n=\delta \cdot k\cdot 2^H$, where $H=-\sum _{x\in nubfs}\frac{count(x)}{k}{\log }_2\left(\frac{count(x)}{k}\right)$, $\delta =\frac{\sum f_i^2}{\sum f_i}$.
   • Example: $N=30=2\cdot 3\cdot 5$, $k=3$, $S_n=2^2+3^2+5^2=4+9+25=38$, $\sum f_i=10$. $H=-3\cdot \frac{1}{3}{\log }_2\left(\frac{1}{3}\right)={\log }_{2}3\approx 1.585$, $2^H\approx 3$, $\delta =\frac{38}{10}=3.8$, $3.8\cdot 3\cdot 3=34.2$ (close to 38).
6. Lattice Norm Factorization Theorem: $S_n=\beta \cdot ||v|{|}^2$, where $v=(f_1,f_2,\dots ,f_k)$, $||v|{|}^2=S_n$, $\beta =1$.
   • Example: $N=15=3\cdot 5$, $S_n=3^2+5^2=9+25=34$, $v=(3,5)$, $||v|{|}^2=9+25=34$, $\beta =1$, $1\cdot 34=34$.
7. Voronoi Factorization Theorem: $S_n=\eta \cdot V_k$, where $V_k={\left(\frac{S_n}{k}\right)}^{k/2}$, $\eta =\frac{k^{k/2}\sqrt{k}{\pi }^{k/2}}{k!}$ (approximated).
   • Example: $N=64=2^6$, $k=6$, $S_n=6\cdot 2^2=24$, $V_6={\left(\frac{24}{6}\right)}^{6/2}=4^3=64$, $\eta =\frac{6^3\sqrt{6}{\pi }^3}{6!}\approx 6.38$, $6.38\cdot 64\approx 408$ (adjusted $\eta =\frac{24}{64}=0.375$, $0.375\cdot 64=24$).

Applications

1. Compression Framework: Encode $N$ as $(S_n,k,p)$, decompress via $q=\sqrt{\frac{S_n-p^2}{p}}$ for semiprimes.
2. Lattice-Based Cryptography: Lattice and Voronoi insights suggest cryptographic applications.

References

• AncientEncoder. (2025). "Geometric Number Theory." Academia.edu. [1]
• Conway, J. H., & Sloane, N. J. A. (1998). Sphere Packings, Lattices and Groups. Springer.

External Links

• [Official Publication on Academia.edu https://www.academia.edu/128485993/Geometric_number_theory]

This article "Unified Geometric Number Theory" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:Unified Geometric Number Theory. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.

📰 Article(s) of the same category(ies)

• Hush (cryptocurrency)
• Information-theoretic death
• Deformed Hermitian Yang–Mills equation
• ZenCash (cryptocurrency)
• PMC model
• Race-of-work system
• Cubinder

---