Some experiments in number theory (1606.05971v1)

Abstract: We experiment with some topics in elementary number theory. For matrices defined by Gaussian primes we observe a circular spectral law for the eigenvalues. We look at matrices defined by Gaussian primes and look at the growth of the determinant, trace. We experiment with various Goldbach conjectures for Gaussian primes, Eisenstein primes, Hurwitz primes or Octavian primes. These conjectures relate with Landau or Bunyakovsky or Andrica type conjectures for rational primes. We also look at some graphs finite simple defined by primes. Finally we look at some spectra of almost periodic pseudo random matrices defined by Diophantine irrational rotations, where fractal spectral phenomena occur. The matrix is the real part of a van der Monde matrix.

• The paper explores various classes of generalized primes and Goldbach-like conjectures within extended number systems through computational experiments.
• The research applies linear algebra techniques like analyzing matrix eigenvalues and uses graph theory on prime configurations to probe structural properties and distributions.
• The experiments provide insights into prime distribution through statistical analysis and reveal connections to classical analytical number theory constants and mathematical physics via fractal spectral patterns.

Insights into "Some experiments in number theory" by Oliver Knill

This paper presents a multitude of exploratory endeavors within number theory, focusing on various classes of primes that extend beyond the traditional rational primes. There are notable discussions around Gaussian primes, Hurwitz primes, Eisenstein integers, and the generalized frameworks these offer for conjectures like Goldbach's.

Exploration of Gaussian Primes

The research starts by considering matrices constructed with Gaussian primes, highlighting an observed statistical regularity in their eigenvalues—an apparent circular spectral distribution. These matrices are analyzed in the context of Goldbach-like conjectures applied to Gaussian primes, a domain where symmetries, such as dihedral ones, lead to bijections with rational primes. Specifically, the research probes density correlations across Gaussian primes by employing statistical methods on rows of eigenvalues, analyzing even and odd distances among these primes, which is suggestive of intricate underlying structures or symmetries.

Goldbach Conjectures for Extended Prime Structures

The text traverses through generalized Goldbach conjectures adapted to different algebraic number systems, notably the realms of Eisenstein and Hurwitz integers, and primes within quaternionic and octonionic frameworks. Such conjectures investigate the sum of two primes in these systems, paralleling Goldbach's original conjecture within the integers. Here, it connects these structures with longstanding issues like Landau's and Bunyakovsky's conjectures, which concern prime density and distribution properties.

Integration of Linear Algebra and Graph Theory

Knill’s experiments extend into linear algebra and graph theory, examining matrices derived from prime configurations and considering their graph representations. These Prime Graphs reveal potential connections between number theoretic conjectures and graph Euler characteristics. Observing the degeneracy or growth of such matrices provides not only insights into prime distributions within complex integer domains but also aligns algebraic and combinatorial concepts.

Bridging to Mathematical Physics via Fractal Spectra

By considering matrices under almost periodic conditions defined by Diophantine irrational rotations, the research identifies fractal-like spectral patterns. These connect to known phenomena in solid state physics related to quasi-periodic structures. Here, further complex transformations of matrix entries reveal potential new pathways to understand determinant growth properties for complex, non-integer dimensions, sparking implications across mathematical physics.

Consistency with Hardy-Littlewood Constants

The relevance of Hardy-Littlewood constants is remarked upon, associating them with prime densities not only over the integers but extended to Gaussian domains. Here, considerable computational work evaluates these constants' precision and convergence within these complex systems, tying these experiments to classical analytical number theory descriptions.

Conclusions and Future Directions

While the paper touches on elaborate responses of number theory in speculative domains, the conjectures extend classical theories into novel territories, urging further theoretical probing, especially within the contexts of divisibility, factorization, and computing eigenvalue distributions in experimentally defined matrices. The interplay with matrices, complex analysis, and almost periodic systems suggests fascinating, uncharted territories for future endeavors within mathematical physics and beyond.

Collectively, Knill's investigations spotlight the rich interconnectedness of algebra, geometry, and analysis within number theory, advocating for a future where computer-assisted explorations further unveil the structures residing within primes of extended complex systems. The paper serves as a vivid illustration of how historical number theoretical quests continue to inspire modern mathematical adventures across various algebraic landscapes.