in our e-book Prime Passages to Paradise
by H. PeterAleff
Volume 1: Patterns of prime distribution
in "polygonal - number pyramids"
1.4.4. The square- to- square "pyramid"
The second texture our scribe noted is that of the successive squares because these tally the growth layers of the number material up to the multiplication of each number by itself. The numbers in those intervals between any two successive squares invite therefore also a methodical comparing.
Accordingly, our busy scribe stacked the so defined number line segments again in symmetrical equilibrium and so obtained the "square-to-square number pyramid".
This version is less steep than the triangular one; you can see its beginning inFigure 7 below and on the second poster.
Here, the cornerstones at the trailing edge are the squares of the layer ordinals n and also the cumulative sums of all the odd numbers up to n, as the pebble figures demonstrated.
Each layer is two entries longer than its predecessor, so the growth increment can be distributed symmetrically without offsetting every other layer from the grid.
The difference between adjacent entries in a column is one less than the difference between the squares of their layer numbers whereas in the triangular- number pyramid, the difference between consecutive entries in either type of column, through the odd or the even layers, is the sum of their two layer numbers.
Unlike masonry blocks, numbers are not subject to gravity, and a number pyramid builder can therefore continue each layer into the "air" beyond the corner stones, simply by repeating there on the same level the successive number line segments that also form the next layer below, and then the next, and so on.
The imagined extensions of the pyramid layers past the "masonry" edges on the same level function here the way good poetry or literature do when these illuminate and clarify threads that are present in the real world but hard to discern there if you donít look beyond its limits. Such works of fiction show you these threads in their imagined and thus more easily focused context.
Similarly, the visually most obvious prime- gatherings in this array appear in that imagined extension. From there, they help us to better discern the straight and curved strings of primes that traverse the number pyramids themselves.
In those extensions, both the triangular- number pyramid and the square- to- square version display several columns that are gaplessly built from primes down to the pyramid edge.
The longest of these columns, at 29 in the first array and at 41 in the second as shown on the posters, continue into the "masonry" with high ratios of primes.
Many other prime- dense columns parallel these solid precursors and continue the prime- gathering structure of both arrays far beyond the limits of any conceivable poster. However, long before these higher-numbered and thus longer columns reach the pyramid edge, each of them gets colonized by a small and slowly growing group of factors that repeat themselves down the column and so create its composite gaps. We will return to the composition of these columns, but for now let me describe their role in the number pyramids.
In both arrays, the entries from each column, prime or not, reappear in diagonals that radiate fore and aft from the column tops at successively shallower slopes.
The angles of these diagonals are symmetrical, but not their heights: with each repetition, the entries from the column get shifted to a higher layer on the right and to a lower one on the left.
Each repetition also separates the column entries with successively greater horizontal intervals. This stretching of the diagonals would make the shallower ones hard to trace, but the compact columns in that imagined airspace behind the pyramids provide convenient guides as to which entries belong to each of the so linked but soon widely dispersed groups.
1.5. Prime-rich edge-parallels
Intricately interwoven between the columns and their diagonal repetitions, you will further find in both posters some lines that run parallel to their pyramid edges and are richer in primes than their surroundings.
Like the column- produced diagonals, these edge- parallels are also reflected across a vertical plumb line from their intersection with the top level. They contain the same entries in both sides of the reflection, again each one a layer higher on the right than on the left. However, unlike those diagonals, the edge- parallels have no column counterparts.
In the triangular- number version, those steep edge- parallels connect numbers in all the pyramid layers without skipping every other one as the columns and their diagonal repetitions do. The entries in the edge parallels alternate therefore between two even and then two odd numbers, like the triangular numbers themselves. They follow the formula Ĺ(n2 + n) + d, and for certain values of d, the odd numbers in their paths include high percentages of primes.
To set these edge parallels visually apart from the above prime columns and their matching diagonals, I joined their numbers in the poster with double lines.
In the square- to- square number pyramid, the odd numbers in the 450 edge- parallels are limited to the even- numbered layers on one side and are repeated in the odd- numbered layers in their reflection on the other. Their form is n2 + f when n is even- numbered, and n2 + 2n + f in the reflection towards the right that repeats the same primes one notch higher in the preceding odd- numbered layer.
Like the numbers from the columns, the entries in these edge- parallels repeat themselves at ever- shallower angles and ever- greater distances with progressive shifts in height. The angles for the initial and most easily traced repetitions of both types are listed in Table 1.
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