Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Emergence of the Quanta Prime Sequence

Moustafa Ibrahim

TL;DR

The paper introduces the Quanta Prime Sequence (QPS) as a unifying framework linking major number-theoretic sequences (Mersenne, Fermat, Fibonacci, Lucas, and Dickson) with harmonic numbers and primality tests such as the Lucas-Lehmer criterion. It develops the Omega and Psi representations that generate new polynomial identities and explicit representations for classical sequences, including a second fundamental theorem and a comprehensive set of prime-emergence results, notably that every $p_{k+1}$ divides a QPS term associated with $p_k$ within an infinite parameter space. It further derives new representations for Chebyshev and Dickson polynomials and reexpressions for Mersenne primes and even perfect numbers, linking these to primality tests and combinatorial identities. The work suggests deep connections to the harmonic series and, potentially, the Riemann Hypothesis, and outlines extensive avenues for future research in prime distribution, Fibonacci/Lucas structures, and cryptographic applications.

Abstract

This paper presents the Quanta Prime Sequence (QPS) and its foundational theorem, showcasing a unique class of polynomials with substantial implications. The study uncovers profound connections between Quanta Prime numbers and essential sequences in number theory and cryptography. The investigation highlights the sequence's contribution to the emergence of new primes and its embodiment of core mathematical constructs, including Mersenne numbers, Fermat numbers, Lucas numbers, Fibonacci numbers, the Chebyshev sequence, and the Dickson sequence. The comprehensive analysis emphasizes the sequence's intrinsic relevance to the Lucas-Lehmer primality test. This research positions the Quanta Prime sequence as a pivotal tool in cryptographic applications, offering novel representations of critical mathematical structures. Additionally, a new result linking the Quanta Prime sequence to the Harmonic series is introduced, hinting at potential progress in understanding the Riemann Hypothesis.

On the Emergence of the Quanta Prime Sequence

TL;DR

The paper introduces the Quanta Prime Sequence (QPS) as a unifying framework linking major number-theoretic sequences (Mersenne, Fermat, Fibonacci, Lucas, and Dickson) with harmonic numbers and primality tests such as the Lucas-Lehmer criterion. It develops the Omega and Psi representations that generate new polynomial identities and explicit representations for classical sequences, including a second fundamental theorem and a comprehensive set of prime-emergence results, notably that every divides a QPS term associated with within an infinite parameter space. It further derives new representations for Chebyshev and Dickson polynomials and reexpressions for Mersenne primes and even perfect numbers, linking these to primality tests and combinatorial identities. The work suggests deep connections to the harmonic series and, potentially, the Riemann Hypothesis, and outlines extensive avenues for future research in prime distribution, Fibonacci/Lucas structures, and cryptographic applications.

Abstract

This paper presents the Quanta Prime Sequence (QPS) and its foundational theorem, showcasing a unique class of polynomials with substantial implications. The study uncovers profound connections between Quanta Prime numbers and essential sequences in number theory and cryptography. The investigation highlights the sequence's contribution to the emergence of new primes and its embodiment of core mathematical constructs, including Mersenne numbers, Fermat numbers, Lucas numbers, Fibonacci numbers, the Chebyshev sequence, and the Dickson sequence. The comprehensive analysis emphasizes the sequence's intrinsic relevance to the Lucas-Lehmer primality test. This research positions the Quanta Prime sequence as a pivotal tool in cryptographic applications, offering novel representations of critical mathematical structures. Additionally, a new result linking the Quanta Prime sequence to the Harmonic series is introduced, hinting at potential progress in understanding the Riemann Hypothesis.

Paper Structure

This paper contains 33 sections, 52 theorems, 171 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Summarized Results from 3 Essential to Our Study
  4. Objective of the Current Paper
  5. Key Discoveries Presented in this Paper
  6. The Quanta Prime sequence
  7. Illustrative Example: Calculating the Quanta Prime Sequence
  8. The representation of in terms of sequence
  9. The Omega space at level
  10. The k-th Prime Number
  11. The Emergence of a New Prime Number
  12. Fascinating Patterns in the Quanta Prime Sequence
  13. The Quanta Prime Sequence Associated with the Point (1, 1)
  14. The Quanta Prime Sequence Associated with the Point (1, 0)
  15. The Quanta Prime Sequence Associated with the Point (1, -1)
  16. ...and 18 more sections

Key Result

Theorem 1

(The First Fundamental Theorem of the Quanta Prime sequence) For any numbers $a,b,\alpha,\beta, n$, $\beta a - \alpha b \neq 0$, we get the following expansion where the coefficients are integers.

Figures (2)

  • Figure 1: The Stable Quanta Prime Sequence at $(1,-2)$
  • Figure 2: Discrete values of $\Psi(1, \sqrt{5}, n)$ for natural values of $n$.

Theorems & Definitions (89)

  • Theorem 1
  • Theorem 2
  • Definition 3.1
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • ...and 79 more