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.
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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 now an unnecessary 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 this said, I offer a quick distillation of what makes a prime and composite in the conventional abstract sense. Note that here the result, dividend, and divisor are considered to be of natural number set: 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). 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, the following is given in the context of dimensional primality and should not be confused with the conventional notion of primes.
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.
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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.
A unit quantity of three satisfies both characteristics and is thus composite.
All unit quantities (≥ two) satisfy both characteristic and are thus composite.
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.
A unit quantity of four satisfies both characteristics and is thus composite.
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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.
A unit quantity of five satisfies the space filling characteristic but not that of equal divisibility and is thus prime.
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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 quantities like 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 composites as described prior, is primality 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 (Euler) conclusive affirmation of primes. Yet if such characteristic were applicable, two would be neither prime nor composite 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?
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 primes 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 nor composite… 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. 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?
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 primality likely never equals, but rather exceeds, that of composites in higher order dimensions. It is then the composites that become illusive. In a number line arrangement (Fig. 3), are higher dimensions populated as a result of lower dimension rationals? Does taking the recursive difference of three dimensional primes validate their existence as it produces a similar pattern to that of conventional primes? (Fig. 4,5,6)
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.
Many thanks to two colleagues and friends, both of extraordinary intellect, character, and perhaps most importantly, curiosity. John Sauvigné and Stephen Schermerhorn.
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 primes:
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 primes:
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 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 three dimensional composites:
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 …
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 nth dimension requires a “knife” of n-1. As such, delineating an embedded n-1 dimension serves as a demarcation of the nth dimension.
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