Prime Number Patterns: Stable Intervals of 10, 20, and Multiples
Prime numbers do not appear randomly. They are components of a system that evidences regularity and sequence.
Prime numbers have been a fundamental focus in mathematics for centuries. Euclid proved their infinitude over 2000 years ago. The challenge with prime numbers is predicting their distribution.
There is actually a clear pattern in the distribution of prime numbers. This pattern solves one of the oldest mysteries in number theory. The pattern connects the fields of mathematics, physics, chemistry, and astronomy.
The System
The system is a circle divided into 9 equal sectors. From the center of the circle, we place natural numbers in order, starting with 1, in a spiral shape expanding from the center to the periphery. The transition to another orbit for all natural numbers will occur at the boundary between sectors 9 and 1 (shown as a red line):
Prime Numbers
We mark the prime numbers in the picture. We see that three sectors of the circle do not contain any prime number. These parts are 3, 6 and 9. They are highlighted in yellow. These sections contain only composite numbers. Primes will never appear in them — to infinity. The only exception is the number 3.
Nikola Tesla had a deep interest in the numbers 3, 6, and 9…
The Spirals
We divide all prime numbers into four groups:
- Prime numbers that end in 1 (pink line): 11, 31, 41, 61, 71, 101, 131 etc.
- Prime numbers that end in 3 (blue line): 3, 13, 23, 43, 53, 73, 83, 103 etc.
- Prime numbers that end in 7 (purple line): 7, 17, 37, 47, 67, 97, 107 etc.
- Prime numbers that end in 9 (green line): 19, 29, 59, 79, 89, 109, 139 etc.
In each of these categories, prime numbers are arranged precisely along the turns of an equiangular spiral. There are a total of four spirals, where the prime numbers appear.
Each group of prime numbers is expected to continue generating in the same manner towards infinity, following their respective spiral paths.
Does it resemble the cosmic web, the large-scale structure of the universe?
Twin Primes
Twin primes (pairs of primes that differ by 2) serve as connecting “bridges” over yellow sections 3, 6 and 9, which always contain only composite numbers.
The “bridges” also connect the neighboring spirals to each other, which looks like they are keeping them stable. All twin pairs are divided into three categories: they end in either 1–3, 7–9 or 9–1. There are no other possible pairs of twins. The only exception is the twin pair 5–7. There are infinitely many twin primes.
The “bridges” between the twin primes are shown as red lines, connecting twin primes: 5–7, 11–13, 17–19, 29–31, 41–43, 71–73, 101–103, 149–151 etc.
How are primes distributed along their spirals?
They seem to be arranged irregularly, but there is a pattern in this irregularity.
When the difference between them is 10, they will always connect the circle sections: section 1 with 2, section 4 with 5, section 7 with 8.
Examples: 7–17, 13–23, 19–29, 31–41, 37–47, 43–53, 73–83, 103–113 etc.
In this case, a prime number moves to another orbital to the next prime number.
When the difference between them is 20, they will always connect the circle sections: section 2 with 4, section 5 with 7, section 8 with 1.
Examples: 11–31, 17–37, 23–43, 41–61, 53–73, 59–79, 83–103, 89–109, 107–127, 131–151, 137–157, 173–193, 179–199, 191–211 etc.
We can see that prime numbers “jump” over the yellow sections since there are no primes there, and there never will be. Interval 20 means, that a prime number jumps to the second orbital to the next prime number:
The Intervals
All four groups of primes are organized uniformly along their spirals: the stable distance between primes is 10 and 20. Let us call them basic intervals.
Apart from these stable distances, there are constant intervals on the spirals of 30, 40, 50, 60, 70, 80 and so on. The smallest interval is always 10. All other intervals are the multiplies of 10, extending into infinity. The intervals can be compared to quantum jumps (how electrons move within an atom).
These “jumps” invariably consist of intervals 10 and 20. The distance between primes on each spiral will always alternate 10–20–10 or 20–10–20 because it follows a sequence in circle sections.
Examples:
Interval 30 (green lines): primes 29–59, 67–97, 71–101, 101–131, 109–139, 149–179, 193–223, 197–227, 199–229, 239–269, 263–293 etc.
Interval 40 (purple line): primes 313–353
Interval 50 (dark blue line): primes 113–163, 383–433
Interval 70 (orange line): primes 331–401
Interval 80 (light blue line): primes 269–349
A high-resolution picture, for zooming in, is available for download here.
As mentioned before, the intervals 10 and 20 will alternate:
30 = 10+20 or 20+10
40 = 10+20+10
50 = 20+10+20
60 = 10+20+10+20 or 20+10+20+10
70 =10+20+10+20+10
80 = 20+10+20+10+20
90 = 10+20+10+20+10+20 or 20+10+20+10+20+10
100 = 10+20+10+20+10+20+10
Intervals that are multiples of 3 (30, 60, 90 and so on) will have two variations of basic intervals’ combination.
All other intervals 40, 50, 70, 80 and so on will have only one basic intervals’ combination.
The intervals indicate on which orbital the next prime number appears:
Interval 10 — the next prime number appears on the 1st orbital from the previous prime number along the spiral.
Interval 20 — on the 2nd orbital from the previous prime number along the spiral.
Interval 30 — on the 3rd orbital from the previous prime number along the spiral.
Interval 40 — on the 4th orbital from the previous prime number along the spiral.
Interval 50 — on the 5th orbital from the previous prime number along the spiral, and so on.
Sustainability of the System
All the natural numbers in each section of the circle follow one principle: from the center to the periphery, all the last digits decrease in order, while all the basic digits increase in order. This sequence will continue infinitely.
Connecting Math, Physics, Chemistry, and Astronomy
The graphical visualization demonstrated a principle of how prime numbers are distributed.
Prime numbers are organised and behave in a way that mirrors electrons in an atom. They are arranged in orbitals around the nucleus. The “nucleus” is the starting point of natural numbers in the center. The intervals between primes on each spiral can be compared to quantum jumps, the discrete steps electrons take when moving between energy levels.
The evidence strongly suggests that the demonstrated pattern remains stable to infinity.
What does this pattern mean for our understanding of the fundamental structure of the universe?
