Induced Dimensional Primality | Steve Wait – February 8, 2024

A Thought Experiment.

Preface –

An adequate accumulation of contextual datum can give rise to patterns, the comprehension of which offers understanding. It is however the data points themselves, their spatial and temporal juncture with lines of inquiry, that are fundamental. Whether these are literal or figurative in occurrence, all answers arrive as consequence of intersection. Having said this, as my enlightenment is my own, perhaps scrutiny will expose faults in the supposition that follows.

A Note to the Reader –

Areas of this thought experiment will require the temporary suspension of your knowledge regarding the foundational principles of prime numbers.

 

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Why the Orthogonal –

As orthogonality plays a role here, I’ll begin with my condensed, anecdotal thoughts on the origins of orthogonal arrangements and representations by humanity at large.

After writing this section, it seems a superfluous justification of something so innate to humanity. It has since been deleted.

The Conventional Context of Composites and Primes –

While the utility of counting numbers is their expressions of reality, the abstractness of mathematics offers alternative context. For certain sets, mathematics demands precision of infinite extent. It’s here that natural, whole, integer, and rational numbers enjoy their usefulness.

With that said, I offer a quick distillation regarding my own interpretation of what constitutes a prime and composite in the conventional abstract sense. Both statements are predicated on the result, dividend, and divisor being of the natural number set:

1. A value is considered prime if it is the result of a dividend that can be operated on by both one and another singular divisor (i.e., itself).
2. A value is considered composite if it is the result of a dividend that can be operated on by both one and multiple other divisors.

 

 

Dimensional Primality

Unless otherwise noted, what follows is given in the context of “dimensional primality” and should NOT be conflated nor confused with our conventional notion of primes and composites. The reader should take great care differentiating my conjecture from well founded mathematical principles. To assist with this, I use “prime-like” and “composite-like” as analogous references to their conventional counterparts.

Attributes:

Natural numbers are used to enumerate the cardinality of quantities occupying space. The arithmetic multiplication of naturals (a subset of wholes which are themselves a subset of integers), as well as the spatial dimensions, are considered to be orthogonal. This, as a geometric arrangement of multiplicand and multiplier (both objects), each along an orthogonal dimension (axis), delineates an array populated by their product.

Characteristics:

Space filling is constituted by a minimum of two objects that also define a dimension’s direction. A sufficient collection of objects are thus capable of a space filling orientation in multiple directions.

Equal divisibility is the division(s) along a dimension between whole objects (≥ two) where the result is an equal quantity. As such, division between objects separates them into groups or whole parts. (Alternatively, division of an object results in a partial object. This represents a subdivision into fractional parts that are not necessarily equal.)

e.g.,

Two as a divisor is one division into two groups or a quantity divided into halves.

Three as a divisor is two divisions into three groups or a quantity divided in thirds.

Four as a divisor is three divisions into four groups or a quantity divided into fourths.

…

 

With the above in mind, a departure from the conventional abstract definition of prime is noted: One as a divisor is superfluous. The resultant of a dividend operated on by a divisor of one is that of the dividend itself (i.e., the definition of division is not met as a separation into parts is not achieved).

Examples:

In one dimension –

The equal divisibility characteristic is applicable to all unit quantities.

A unit quantity of one is dimensionless; it satisfies neither characteristic.

Two is the first unit quantity capable of constituting one dimension; it satisfies both characteristics and is thus “composite-like”.

A unit quantity of three satisfies both characteristics and is thus “composite-like”.

All unit quantities (≥ two) satisfy both characteristic and are thus “composite-like”.

…

 

In two dimensions –

The equal divisibility characteristic substantiates composite unit quantities.

A unit quantity of one is dimensionless; it satisfies neither characteristic.

A unit quantity of two is capable only of constituting one dimension.

Three is the first unit quantity capable of constituting two dimensions; it satisfies the space filling characteristic but not that of equal divisibility and is thus “prime-like”.

A unit quantity of four satisfies both characteristics and is thus “composite-like”.

…

 

 In three dimensions –

The equal divisibility characteristic substantiates composite unit quantities.

A unit quantity of one is dimensionless; it satisfies neither characteristic.

A unit quantity of two is capable only of constituting one dimension.

A unit quantity of three is capable only of constituting two dimensions.

Four is the first unit quantity capable of constituting three dimensions; it satisfies the space filling characteristic but not that of equal divisibility and is thus “prime-like”.

A unit quantity of five satisfies the space filling characteristic but not that of equal divisibility and is thus “prime-like”.

…

 

Note here that for three dimensions, an “irregular quantity” could be defined as any quantity that, while capable of contiguously populating a lattice arrangement in three orthogonal directions, is not equally divisible in each (Fig. 2., of original draft). Eight then, is the first quantity that enjoys both the space filling characteristic and that of being equally divisible in each direction. As such, a “regular quantity” could be defined as any quantity that can fulfill a regular lattice arrangement in three orthogonal directions and is thereby equally divisible in each. Put another way, regular quantities are the product of three natural numbers, all greater than one.

An intuitive extension of the aforementioned resides in the semi-abstractness of two dimensions where the tiling of a plane in orthogonal directions is possible. In this scenario, the quantities of one and two lack any ability to fill in two orthogonal directions. If we assume that for one dimension (where all quantities are equally divisible) the first linear filling quantity in one direction is two, and that for three dimensions, the first space filling quantity in three directions is four, it will seem logical that for two dimensions, it’s first plane filling quantity is three. And indeed, three does satisfy the filling characteristic but does not possess the equal divisibility characteristic. Applying the prior definitions in this two dimensional construct reveals that not only are “composite-like” quantities such as four, six, eight, nine, etc. plane filling, but they can be arranged with equal divisibility in both directions (Fig. 1., of original draft).

The Many Questions –

Using this spatial definition for “composite-like” quantities as described prior, are “prime-like” quantities implicit for the remainder? Generally, yes. Specifically… not so fast. Foregoing the debate on the “primality” of one, it is the quantity of two that presents itself as unreconcilable in this context. Two’s absence of the filling characteristic, as with one, two, and three in three dimensions, brings pause. Obviously, the equal divisibility characteristic, sanctioning the exclusion of two from “primality”, is less than a palatable notion in the conventional sense of primes. In the pure abstract of mathematics, the idea runs up hard against the Fundamental Theorem of Arithmetic and afoul of Riemann’s zeta function as conclusive affirmation of primes. Yet if such characteristic were applicable, two would be neither “prime-like” nor “composite-like” in this two dimensional construct. It is the seeming absurdness of just this that begs the question: What then of the quantities one, two, and three in three dimensions? They are neither space filling nor equally divisible by definition. What then, are they? Does any of this intersect with the Riemann Hypothesis or perhaps the Liouville Function?

Having said this, regarding the primality of two and the Fundamental Theorem of Arithmetic, it seems only their mutually enabling relationship provides sustenance. How then can we fulfill the uniqueness characteristic of the Fundamental Theorem of Arithmetic without the number two? Perhaps it requires a “seed” from a lower dimension? (Fig. 3) Perhaps the applicability of the Fundamental Theorem of Arithmetic across spatial domains necessitates quantities that are absent a given dimensional arrangement’s filling requirement. To that end, as these quantities advance successively through each dimensional arrangement, they must function as “prime-like” regardless of their lower order “primality”. What then are those necessary quantities? Is there then a third species of quantity… one that is neither “prime-like” nor “composite-like”… one that is necessary to fulfill the Fundamental Theorem of Arithmetic in a context of “dimensional primality”? Does another circumstance exist where both conditions can be true? If objects are placed in a dimensional arrangement such that they fill space and are equally divisible in each dimension, then they are “composite-like”. When extending into three dimensions, it would seem the Fundamental Theorem of Arithmetic, in a physical sense, could be preserved. Conceptually, could Riemann’s affirmation possibly hold in three dimensions less two (and three) as “prime-like”?

Extending beyond three dimensions seems suggestive of equal divisibility becoming less and less likely for reasons similar to that referred to in the later parts of A Mutual Division of Space . As such, the density of “prime-like” quantities seems likely to never equal, but rather exceed, that of “composite-like” quantities in higher order dimensions. If so, it is then the “composite-like” quantities that become illusive. In the number line arrangement (Fig. 3) of quantities in each dimension up to and including four, are higher dimension quantities subsequently populated as a result of lower dimension rational numbers? And cautiously, does taking the recursive difference (see Integer Sequence Fingerprints via Recursive Difference) of three dimensional “prime-like” quantities lend any validation to their existence as it produces a pattern similar to that of the conventional primes?

At best, all the aforementioned questions are unclear and beg further exploration.

An Intermission For Now –

Couldn’t most all these questions be rendered moot by withholding consideration of the space filling characteristic, relying solely on the divisibility characteristic to classify irregular quantities? That seems a dodge. And so, perhaps reconciliation simply resides in the forest… behind all those trees.

 

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Initial Sequence Values –

Those satisfying the characteristics of conventional primes:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541 …

 

Those satisfying the characteristics of two dimensional “prime-like” quantities:

3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547…

 

Those satisfying the characteristics of three dimensional “prime-like” quantities:

4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118, 119, 121, 122, 123, 127, 129, 131, 133, 134, 137, 139, 141, 142, 143, 145, 146, 149, 151, 155, 157, 158, 159, 161, 163, 166, 167, 169, 173, 177, 178, 179, 181, 183, 185 …

 

Those satisfying the characteristics of conventional composites:

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 128, 129, 130, 132, 133 …

 

Those satisfying the characteristics of two dimensional “composite-like” quantities:

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 128, 129, 130, 132, 133 …

 

Those satisfying the characteristics of three dimensional “composite-like” quantities:

8, 12, 16, 18, 20, 24, 27, 28, 30, 32, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 116, 117, 120, 124, 125, 126, 128, 130, 132, 135, 136, 138, 140, 144, 147, 148, 150, 152, 153, 154, 156, 160, 162, 164, 165, 168, 170, 171, 172, 174, 175, 176, 180, 182, 184, 186, 188, 189, 190, 192, 195, 196, 198, 200, 204, 207, 208, 210, 212, 216, 220, 222, 224 …

 

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A Quick Sidebar –

As a sidebar to this thought experiment, it has caused me to consider the divisibility of a dimension. It would seem to divide something of n^th dimension requires a “knife” of n-1. As such, delineating an embedded n-1 dimension serves as a demarcation of the n^th dimension.

 

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