Barry Dalgarno
The sequence of the prime numbers is finely tuned from the perspective of symmetry. This was first shown in 2009 (Ref. 1). In that paper, which describes the structure of minimum vertices, the natural numbers from 2 onwards are positioned on a grid, with the primes forming the axis. Positioning is such that each number is connected to the next in a continuous network, with no step ever retracing its previous path. This type of geometric entanglement of the prime number intervals can result in the formation of square symmetry. Although the internal structure with respect to the prime intervals is asymmetric, the perimeter is symmetric. For an introduction to geometric entanglement, see reference 1. In the table below, the prime number range relates to the sequence of consecutive primes, positioned along the square diagonal (prime axis).
Size Prime Number Range Interval Sequence
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5 x 5 479 to 503 8-4-8-4
7 x 7 3019 to 3067 4-14-4-8-12-6
9 x 9 35537 to 35617 6-26-4-18-2-4-6-14
11 x 11 186107 to 186227 6-6-30-8-4-2-24-4-20-16
13 x 13 1228399 to 1228567 30-12-16-2-30-12-18-18-4-2-4-20
15 x 15 7504787 to 7505011 2-10-8-10-24-56-4-8-48-4-6-12-14-18
17 x 17 50033869 to 50034157 22-32-4-72-14-16-8-4-32-6-28-6-8-12-6-18
19 x 19 314669617 to 314669977 52-18-2-10-18-14-10-18-32-28-2-58-12-2-28-18-20-18
21 x 21 2000058251 to 2000058691 20-18-4-44-12-34-8-6-16-68-10-2-28-26-12-22-2-64-26-18
23 x 23 12447734281 to 12447734809 42-78-36-6-34-2-4-18-38-70-14-24-12-12-16-36-14-6-28-20-10-8
25 x 25 72004935889 to 72004936513 40-42-26-24-36-10-12-42-2-46-14-6-18-64-6-26-18-30-18-22-6-80-34-2
27 x 27 532048248713 to 532048249441 14-22-2-18-22-26-16-30-6-2-60-6-6-54-12-4-8-6-12-10-20-54-54-22-122-120
29 x 29 4606127184359 to 4606127185199 32-22-54-12-2-18-40-14-66-22-6-2-4-8-10-68-138-72-10-14-34-54-2-6-22-32-70-6
31 x 31 80000000138609 to 80000000139569 14-24-52-12-2-96-4-14-70-12-8-34-68-70-30-8-6-114-24-70-2-4-44-48-48-4-6-20-40-12
33 x 33 510000000001763 to 510000000002851 18-60-20-12-28-12-26-90-10-8-42-34-74-90-6-10-96-8-4-36-12-44-108-18-64-48-6-60-6-24-12-2
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 (2014). Prime Number Interval Squares
https://sites.google.com/site/geometryoftheprimes/prime-number-interval-squares-2