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Barry Dalgarno
4 April 2024
Dedicated to my mother Ruth Dalgarno
Geometric entanglement of the prime number intervals can result in the formation of square symmetry. The purpose of this paper is not to provide an exhaustive list of prime number interval square speciation, but to show some examples of the various types. There are five types of speciation :
Type 1 : Prime Number Range
Type 2 : Interval Sequence
Type 3 : Interval Permutation
Type 4 : Interval Geometry
Type 5 : Interval Pathway
Prime Number Range Speciation
The first type refers to where a given size of square can be formed over more than one sequence of consecutive prime numbers. This can be seen for example with reference to the first three prime number interval squares of size 5 x 5. The first square occurs from prime 479 to 503. The second square occurs within the prime range 1559 to 1583. The third square occurs within the prime range 8087 to 8111. (see reference 3).
Interval Sequence Speciation
The second type of speciation refers to where a given size of square can be formed from more than one interval sequence. This can be seen for example with reference to the third square of size 5 x 5, as the interval sequence (2-4-8-10) differs from that of the first two squares of size 5 x 5 (8-4-8-4) (again see reference 3).
The remaining three types of speciation will be shown with reference to the following 13 x 13 prime number interval square as an example:
Consecutive prime numbers 1228399 to 1228567.
Interval sequence 30-12-16-2-30-12-18-18-4-2-4-20.
In the figures shown, the unfilled circles along the square diagonal (horizontal) represent consecutive prime numbers, and the smaller filled circles represent the non-primes occupying the prime intervals.
Interval Permutation Speciation
This type of speciation arises where, for a given interval sequence, there is at least one alternative permutation of the intervals that still fits together to form a completed square structure. In the case of the 13 x 13 square referred to above, the two 12-intervals (shown in green), the two 18-intervals and the two 30-intervals can each be swapped over with respect to the prime axis. In the first figure below, the first 12-interval, first 18-interval and second 30-interval are geometrically entangled with three other intervals below the prime number axis. In the second figure, the first 12-interval, first 18-interval and second 30-interval are geometrically entangled with three other intervals above the prime number axis.
Interval Geometry Speciation
For a given interval, an alternative spatial geometry is sometimes possible. By way of example, the second 30-interval in the first figure above can adopt a number of different geometric arrangements, one of which is shown in the second figure below (in red).
Interval Pathway Speciation
Positioning of the natural numbers on the grid is such that each number is connected to the next in a continuous network, with no step ever retracing its previous path. Whilst always maintaining sequential numerical order, it is sometimes possible to trace alternative pathways through the numbers for a given interval. In the figures below, the first 12-interval is shown (in green) with different pathways.
References
1. B. Dalgarno (2009). A New Law of Prime Numbers Based on an Infinite Structure of Sequential Triangular Symmetry from Finely-Tuned Geometric Entanglement of the Prime Intervals
https://sites.google.com/site/geometryoftheprimes/
2. B. Dalgarno (2024). A 13 x 13 Prime Number Interval Square
https://sites.google.com/site/geometryoftheprimes/a-13-x-13-prime-number-interval-square
3. B. Dalgarno (2014). Prime Number Interval Squares
https://sites.google.com/site/geometryoftheprimes/prime-number-interval-squares-2
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