Barry Dalgarno

18 December 2009

# Dedicated to my mother Ruth Dalgarno

Abstract

This paper concerns the discovery of a new law of prime numbers, based on finely-tuned geometric entanglement of the prime number intervals, and an infinite structure of sequential triangular symmetry. A deeply rooted geometry has long been sought to explain the accumulation of the primes. Geometric entanglement of the intervals is an operation that reveals the symmetry underlying the step-by-step distribution of the primes.

### Introduction

The history of research into the prime numbers has been a long and colourful one. Many mathematicians have contributed in their own ways, and it is not possible to mention all of these intrepid explorers. Their work has helped to lay the foundations for the position we are fortunate to find ourselves in today. One of the first pioneers was Adrien-Marie Legendre (1752 - 1833). His important work was instrumental in securing the necessary knowledge on prime distribution. Carl Freidrich Gauss played a decisive role in laying the groundwork, mainly by way of the Prime Number Theorem. This was subsequently proved by Jaques Hadamard, and also by Charles de la Vallee-Poussin.

In 1859, announcement of the Riemann Hypothesis was an epoch-making event. It was the discovery of a beautifully simple law that explained the step-by-step accumulation of the prime numbers. Professor Bernhard Riemann was a German mathematician, whose brilliant work both inspired and galvanised future research. His untimely death was a great loss to mathematics and theoretical physics. Exploration continued, and Thomas Stieltjes carried out remarkable research into Riemann's hypothesis. There was every prospect that his deep insight would have delivered a proof. Unfortunately, Stieltjes passed away before he could submit a manuscript for publication. This was another great loss to the world of mathematics. And so the quest continued to the present. A deeply rooted geometry has long been sought to explain the distribution of the primes.

Method and Results

The natural numbers from 2 onwards are positioned on a grid, with the primes forming the axis. Positioning is such that each number is connected to the next in a continuous network, with no step ever retracing its previous path. Employing full degrees of freedom in terms of forward and backward-sloping diagonal path propagation both above and below the axis, leads to the geometry. There is an infinite set of possible geometric arrangements. Consider the infinite subset in which the lattice tessellation leaves no discontinuity in the perimeter. For the two smallest intervals (2 and 4), it is always possible to trace a single triangle from the vertex to the prime axis. Every prime interval greater than 4 is of sufficient height and reach to encounter another interval template, so that a composite triangle can be traced from the vertex to the prime axis. Within this subset, two specific subsets are of particular interest. They are listed according to their respective defining property.

1. Minimum ratio of perimeter vertices to prime-count (referred to simply as The Structure of Minimum Vertices).

2. Maximum ratio of perimeter vertices to prime-count (referred to simply as The Structure of Maximum Vertices).

Now consider the subset in which the ratio of perimeter vertices to prime-count is a minimum (the first subset referred to in the above list). The operation of geometric entanglement (xi) is addressed later in this paper. Various arrangements are possible with respect to rotation of the structure by 180 degrees, at primes where there is no geometric restriction. For example, rotation at primes 5, 11 and 17 would fulfil this requirement, whereas rotation around primes 7, 13, 19 and 23 would not. Any one of these allowed rotational variations will be referred to as the structure of minimum vertices.

The lattice is composed of an infinitely rich and diverse sequence of overlapping and non-overlapping triangles, with the base of each triangle residing on the prime number axis. The whole structure is the result of an equally diverse mosaic of prime interval templates. Depending on the overlap, a given prime interval can also contribute to the formation of a neighbouring triangle. For each rotational variation of the lattice, the aperiodic series of triangles gives the overall impression of superimposed waves, albeit of triangular waveform. Imagine a 'linear kaleidoscope' of triangular facets. One that never repeats the same facet sequence. The structure is described here as resembling a 'reverse fractal'. Although growth of the set of triangles is infinite, the fractal-like appearance of the perimeter shows a diminishing ratio of vertices to prime-count N, and the ratio decreases towards zero at an ever slowing rate as the 2-dimensional lattice propagates.

At first glance, the sequence ranging from single template constructs, to composite triangles of greater size and internal complexity appears to be irregular. However, within this apparent irregularity has lay hidden the very information from which the law of the primes is derived. It is encoded within each and every triangle of the sequence, in the form of geometric entanglement of the intervals. Entanglement (xi) is the number of prime interval templates per triangle. A value of one refers to a triangle with respect to the prime axis and a perimeter vertex, formed from a single template. A value of two refers to a composite triangle formed from the entanglement of two templates, and so on.

Detail of structure. Consecutive prime numbers 1289 to 1429 (axis of unfilled circles).

Filled circles represent non-primes occupying the prime intervals.

The ratio of cumulative average entanglement (xi mean) to the natural logarithm (base e) of cumulative average prime-rank (NT mean, relating to completed triangles with respect to the prime axis) converges on 0.5. The corresponding law is as follows.

In the limit when NT approaches infinity :

Conclusion

Geometric entanglement of prime number intervals is an operation that reveals the symmetry underlying the step-by-step distribution of the primes. The intervals can be seen as a manifestation of geometric harmony. The sequence of prime intervals is so finely tuned that by following the encoded propagation path of minimum vertices, geometric harmony is forever approached, and only realised at infinity.

Bibliography

Rockmore, D. (2005). Stalking the Riemann Hypothesis: The Quest to Find the Hidden Law of Prime Numbers. Jonathan Cape, London.

du Sautoy, M. (2004). The Music of the Primes. Perennial, New York.

Maor. E. (1994). e: The Story of a Number. Princeton University Press, Princeton, N.J.

Goldstein, L.J. (1973). "A History of the Prime Number Theorem". American Mathematical Monthly 80, no. 6, pages 599-615.

Tenenbaum, G. and Mendes-France, M. (trans. Spain, P.G.) (2000). The Prime Numbers and Their Distribution. American Mathematical Society, Providence, R.I.

Edward, H. (2001). Riemann's Zeta Function. Dover Publications, Mineola, N.Y.

References

Ruth Dalgarno - A Life Remembered. https://sites.google.com/view/ruth-dalgarno/home

Dalgarno, B. (2011). Geometric Entanglement of Prime Number Intervals: An Operation that Reveals the Symmetry Underlying the Distribution of the Primes.