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Twin primes and the $3$-sphere

Samuel A. Hambleton

Abstract

We investigate the group of points of the $3$-sphere modulo a prime, point out connections to other known groups and the Chebyshev polynomials, and show that there is an infinite series which converges if and only if there are finitely many pairs of twin primes. Hence to prove that that the series diverges is to prove the twin prime conjecture.

Twin primes and the $3$-sphere

Abstract

We investigate the group of points of the -sphere modulo a prime, point out connections to other known groups and the Chebyshev polynomials, and show that there is an infinite series which converges if and only if there are finitely many pairs of twin primes. Hence to prove that that the series diverges is to prove the twin prime conjecture.

Paper Structure

This paper contains 5 sections, 10 theorems, 54 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Primes and the group modulo $n$
  3. Twin primes
  4. Circles, spheres, and calculation of $n X$
  5. Acknowledgments

Key Result

Proposition 1.1

Let be points of the $3$-sphere $\mathcal{S}$ with coordinates in the commutative ring with unity $\mathcal{R}$ such that the imaginary number $i \not\in \mathcal{R}$. Define the maps where $\text{SU}_2 \left( \mathcal{R} \right)$ denotes the special unitary group of degree $2$ with entries in the ring $\mathcal{R}[i]$ and $\text{SO}_4( \mathcal{R} )$ is the special orthogonal group of order $4$

Figures (2)

  • Figure 1: The group law on the unit circle.
  • Figure 2: The model of $\tau (2, m)$ given by \ref{['tau2mod']}.

Theorems & Definitions (24)

  • Proposition 1.1
  • proof
  • Remark 1.1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.1
  • proof
  • ...and 14 more