Anyone interested in mathematics has likely heard about prime numbers, but almost no one analyzes these numbers in terms of their products and squares. As you may have already read in the section PRIME NUMBERS CODE, all prime numbers (except for 2 and 3) always belong to the sets of numbers:
\( \ 6k±1 \mid k \in \mathbb{N}, k \geq 1, \lim_{k \to \infty} \)
These formulas, with additional appropriate assumptions, eliminate \( 73\frac{1}{3}\% \) of all natural numbers that cannot be prime numbers. As a strong argument, I will add that 50% of the numbers in the Fibonacci sequence also belong to the formulas 6k±1. More details can be found in the section FIBONACCI SEQUENCE. To avoid misunderstandings, I emphasize that here I analyze prime and near-primes numbers starting from the number 5.
As I mentioned at the beginning, all prime numbers (starting from the number 5) always appear in the set of numbers derived from the formulas 6k±1. Of course, to make it less straightforward, these sets also include the products and squares of prime numbers (starting from 5), which are near-primes numbers. Near-primes numbers also always occur exclusively in these two sets and take the form:
\( \ (6k + 1)(6k-1) \mid k \in \mathbb{N}, k \geq 1, \lim_{k \to \infty} \)
\( \ (6k + 1)(6k + 1) \mid k \in \mathbb{N}, k \geq 1, \lim_{k \to \infty} \)
\( \ (6k-1)(6k-1) \mid k \in \mathbb{N}, k \geq 1, \lim_{k \to \infty} \)
Here are a few of the initial near-primes numbers listed:
25, 35, 49, 55, 65, 77, 85, 91, 95, 115, 119, 121, 125, 133, 143, 145, 155, 161, 169…
Every computer scientist will notice that near-primes numbers are nothing more than the n values of the public key in RSA encryption.
Of course, one might hastily conclude that near-primes numbers are not particularly remarkable. However, at this point, I will reference the discovery of a graphical method that demonstrates certain unexplained regularities in the distribution of prime numbers, a method previously thought to be absent. This approach was proposed by the Polish mathematician Stanisław Ulam in 1963.
Ulam Spiral vs. Near-Primes Numbers
Let us compare the Ulam spiral with near-primes numbers marked on the spiral. We observe a very similar arrangement of near-primes numbers to that in the Ulam spiral (where prime numbers are highlighted). The only noticeable difference is the greater quantity of near-primes numbers compared to prime numbers, as shown below:
Let’s take it a step further and overlay the typical Ulam spiral with prime numbers onto the spiral with near-primes numbers. The result can be seen below.
What we observe is a symmetrical arrangement of prime and near-primes numbers, as seen below. These numbers complement each other, forming a repetitive structure. Therefore, we must conclude that the missing link in understanding the code of prime numbers is, in fact, near-primes numbers, which are essentially an inseparable consequence of the existence of prime numbers and their product. The occurrence of prime numbers is inherently linked to the occurrence of near-primes numbers.
An important aspect of this discovery is that we are dealing here with a situation in which, based on relative symmetry and a repeating graphical pattern, we can not only more easily predict the future occurrence of prime numbers, but we also have an easier possibility of factoring near-primes numbers that are values of n in the RSA algorithm. Details soon.
In the PRIME NUMBERS CODE section, the issue is described in detail, explaining that based on the above information, there is a formula for prime numbers.
The occurrence of, among other things, the zero points of the Riemann function and the distribution of energy levels in the nuclei of heavy atoms may be strongly influenced by near-primes numbers, rather than, as commonly believed, only by prime numbers. Therefore, the scientific community should consider near-primes numbers as equally important as prime numbers.
Below I present a spiral created only by numbers from the patterns 6k±1 where prime numbers are marked in black. As we can see below, prime numbers again create some unexplained regularities in their distribution, arranging themselves in lines.
As a curiosity, we can also exclude from the set of near-prime numbers the values that are multiples of the number “5“, then we will get the set:
49, 77, 91, 119, 121, 133, 143, 161, 169, …
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