The Prime Echo
A Complex Visual Pattern In Prime Numbers
And
A Potential Tool for Plotting Primes and Coding
By Christopher R. Baker
A new type of Prime number pattern has been discovered using a drawing compass and number line on paper. While sketching a separate project, and at the same time having explored Prime, Fibonacci, Fractal and other number patterns a number sequence was realized. It was noticed that on either side of Thirty were two Prime numbers Twenty-Nine and Thirty-One and that these numbers totaled to Sixty. Then The Prime numbers Twenty-Three and Thirty-Seven also arced around Thirty and totaled to Sixty. One can take any number and find the equidistant bracketing numbers will equal twice the original number. However, around the number Thirty, the Prime numbers from one side of the arc land on Prime numbers on the opposite side more often than not. This Prime number concentric alignment seemed to be uniquely associated to the number Thirty. Soon after, negative Thirty was also included as a concentric focal point for this ring pattern, a first clue to a bigger discovery. Prime numbers resonate from these central points in corresponding pairs. Limitations with paper and compass provided only a narrow view of this pattern. Using computer programs to draw up the pattern on a larger number graph line helped to reveal this extensive multi-layering. Studying the graph models led to some extraordinary conclusions.
The first page image shows concentric rings in Blue on a barely visible red number line (zoomed out view). This is the basic pattern of Prime numbers arcing in pairs. These arcs have corresponding Primes on both sides of the number line. This pattern looks like a sound wave pattern at first glance and has been dubbed the Prime Echo. Limitations in the 3D modeling program used to create this art reduced the detail in order to fit this full arc image. Gallery 2 shows the (zoomed in) detailed graph line of Prime numbers but the art doesn't provide the full arcs linking each side. The green numbers provide corresponding left and right sides of these arcs.
In some instances, (Gallery 3) a Prime number on the left or right side of center does not have a corresponding Prime on the other side, but the arc fits and enhances the overall pattern structure despite this. These arcs land on composites of Prime numbers on their opposite side. Examples: -49 + 109 = 60, however -49 is not prime. It is a composite; 7 x 7 = 49. -77 + 137 = 60. 77 is a composite of 7 x 11, two Primes. The rings are indicated with colors Blue, Red and Purple for contrast. Blue has Two Prime numbers the basis of the discovery. Red and Purple rings have just One Prime number. Green numbers again match the lines corresponding to the left and right sides of the rings for reference. The black numbers are the diameter of each ring. Listing these numbers led to realizing these arcs were the result of Prime radius measurements; another piece of the discovery.
Gallery 4 shows the overall pattern including the one-sided Primes, the red and purple lines. The pattern is visually simple to perceive but the nature of the pattern is complex and holds many sub-pattern sets. This new type of Prime number pattern is identified as a visually concentric construct, based around the number Thirty, on a number line graph with mathematical properties in the patterning. The pattern consists of rings that fall on Prime numbers on a number line of whole integers in the positive and negative directions centered around Thirty. These complex sub-patterns can be further complicated by copying the same set of rings and pasting over negative Thirty.
A key code is provided with the Long Graph art in Image Three. This was the most important image, before Image Five was created. The Red Number Line is the Prime Number sequence up to positive 400, with the tens place marks above the red graph line.The blue lines indicate the arc rings that have corresponding Prime numbers on both sides of the ring. The highest concentration of blue is nearest the center and diminishes as the wave spreads. The purple rings denote the lines that have Prime numbers only on the right side of the ring but not on the left. The red rings indicate the Prime numbers on the left of center only. Purple and red lines have only one Prime with a corresponding composite number opposite. These lines increase their ratio relative to the blue lines as they spread. Their are Forty-Nine blue lines up to 400. The red and purple lines also total Forty-Nine up to 400. The green numbers serve to identify the corresponding left and right sides of each ring, since the art does not provide the complete arcs in the image. The black numbers are the diameter of each ring. These diameters when divided by two are mostly prime number radii with some being composite numbers. Also note, the images show occasionally void areas. These represent parts of the expanding patterning as well.
This document is meant as a brief introduction for the Prime Echo pattern. Mathematicians and Prime number enthusiasts are left to further explore the higher numbers that this attached art promises will reveal greater values. Some examples of this patterning for study include; (1) mathematically noting the visual rhythm of the rings to design a computer model that could extrapolate and predict the pattern in extremely high numbers, (2) attempting to generate sounds that match these patterns, (3) finding occurrences in nature and machinery of sounds or patterns generated with these frequencies, (4) identifying the many sub-frequencies within the pattern.
One of these sub-frequency patterns coupled with other clues led to a declarative statement. The image in Gallery Five should be the proof. It was noticed on the number line, that a sub-pattern of ring lines land closest at magnetic Compass points (30, 60, 90, 120, etc.) These lines have the closest proximity; only Two integers apart, (29-31, 59-61 etc.). Although, this pattern does skip at 360, lacking a ring on one side. The sequence then continues 390, 420... and so on. Does this skip have a rhythm in the expansion?
Of important note, that pattern led to realizing the spacing of ring lines is equidistant from these Compass numbers, in both directions repeating at prime distances. The Prime numbers overlap in waves traveling in both directions. This realization and the notice of the rings' radii being Prime, prompted the pasting of the ring array over numerous Compass numbers. (See Gallery 5) In the first Four images the waves spread and expand as the numbers increase, making the pattern harder to perceive and predict. However, using the same set of rings shifted to other multiples of Thirty causes these rings to overlap and identify Prime numbers. These details were discovered during this writing and lead to the following declaration.
This document declares that there is now a pattern and a model to work from in which Prime numbers may be predicted and plotted. A graphic program could plot in analogous image form and numerical programs could plot digitally.
The resulting pattern is an offset mirroring of numbers that could be used for encryption which should be extremely confounding, if not impossible to decrypt because one number can imply the existence of another number without pointing to it. And the start point can be shifted anywhere along the graph line then buried within the multitude of overlaying ring sets. Knowledge of the Prime pattern could increase computational mathematics and computer capabilities.
The search for an ultimate defining rule for the sequence of Prime numbers may be this pattern. Layering these Prime numbers at an offset frequency of multiples of Thirty, using Prime numbers to define the order, plots their positions all along the number line at intersecting points.
I leave it to experts to study further.
For questions, use e-mail. Christopher@engineer.com
Christopher R. Baker
