Prime Number Interval Squares

Barry Dalgarno

Originally published on 20 September 2014 under the title : Square Symmetry from Geometric Entanglement of the Prime Number Intervals. 

Dedicated to my mother Ruth 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. The first square symmetry unit is shown below. It is a 5 x 5 square formed from the prime interval sequence 8-4-8-4. It occurs from prime 479 to 503. For brevity, square symmetry units like this will be referred to simply as prime number interval squares. In the figure 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. 

 As we move along the prime axis, the next 5 x 5 prime number interval square occurs from prime 1559 to 1583, and it is formed from a similar sequence of intervals (8-4-8-4). Moving further along the axis we encounter the third 5 x 5 prime number interval square within the prime range 8087 to 8111. Here the interval sequence is different (2-4-8-10).  

Examples of larger prime number interval squares are shown here.  

A 7 x 7 square. Consecutive prime numbers 3019 to 3067. Interval sequence 4-14-4-8-12-6.

A 9 x 9 square. Consecutive prime numbers 35537 to 35617. Interval sequence 6-26-4-18-2-4-6-14.  

An 11 x 11 square. Consecutive prime numbers 186107 to 186227. Interval sequence 6-6-30-8-4-2-24-4-20-16.

Conjecture 1

There is an infinite set of prime number interval squares.

Conjecture 2

There exists at least one prime number interval square for each and every square size, from 5 x 5 onwards.

Starting with 5 x 5 :

5 x 5 , 7 x 7 , 9 x 9 ...         to n x n

where n covers all the odd integers from 5 to infinity.

References

Ruth Dalgarno - A Life Remembered.

 https://sites.google.com/view/ruth-dalgarno/home

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/