Abstract

The prime orbital sequence (SOP) puts order in the set of prime numbers.
It is made up of I_n terms, inside which 1, 2 or more prime numbers are stored, which we will say belong to it..
So for example 13 is ph » brother prime» of 7, and the two are the brother prime numbers that constitute I₆ = 91
In the same way 41 and 271 are ph of 11111 = 1₅ = I₅.
The work calculates the In by means of a somewhat complicated algorithm, and for this it studies the properties of the giant numbers (NG).
They are numbers with structure, or derived from simple operations with structured numbers.
Structured numbers are those that can be represented by an expression similar to that used in chemistry for molecules..
For example:
I₅₀ = 9₅0₅9₅0₄1 that carries within itself three sibling prime numbers, 251* 5051* 78875943472201.
I₃₄₂ = (10₃9₅89₃0₅)₂10₃9₅89₃0₆9₅89₃0₅10₃9₅89₃0₅10₂1.
Of many of them we will not know if they have ph inside, or if they themselves are prime numbers.
There are simpler ones like I₁₀ = 9091 that does not have prime brother numbers inside since he is a prime.
Or I₁₃ = 1₁₃ that carries within 53*79*265371653.
I₂ = 11, I₃ = 3* 37, I₄ = 101, I₁₂ = 9901; I₂₄ = 9₄0₃1 that do not have
phs, since they are two precious prime numbers. . . etc.
The work also includes the properties of cyclic numbers that are prime
∀p ∈1_p−1.
It informs how to find prime numbers of n digits, no matter how high it is n.
Easy to express subsequences are written but with terms as large as we want.
We treat various algorithms such as the IIn and the “algorithm del anterior”.
And I include properties of the primes and theorems about them, such as the “repeated numbers theorem” and the “continuity»
theorem”:a_p − a^p ≡ 0(p. Or the “superior prime «theorem.

Introduction

The Orbital Prime Number Sequence is a work of years, years trying to make its terms easier to calculate.
From the various properties derived from this effort, I hope to open new lines of research in this exciting field.
This work has been created without looking at any other author, because it is a great pleasure that everything comes out of your inner self.
I only use Fermat’s petit theorem, because I realized that it was the same one that I used (I used ,10 as coprime, which is always co-prime with 2 and 5).
The mathematics used in this work are elementary and can be understood by anyone, who spends a little time on this work.
I have given examples of everything I say. You just have to check it out. It is a work, more research than a mathematical theory.
The SOP is a sequence that stores all the prime numbers.
It is like a train, in each wagon there are several prime numbers, which I call “brother primes”.
For example, the passengers of car 13 are 53, 79 and 265371653. These are three brothers primes. We’ll see why further down.
The passengers of wagon 10 are divisors of term 10 of the SOP, which is 9091, but it turns out that this is the only passenger and does not carry anyone else, because it ∈ IP.
99999000009999900001 is term 50 of the SOP, and includes, as passengers, prime brothers 251, 5051 and 78875943472201.
I don’t know who the passengers in car 100 are, but I do know that they divide 9999999999000000000099999999990000000001, which is term 100 of the SOP.

1. Formulation and nomenclature

Euler is credited with the phrase: «Mathematicians have tried in vain to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery that the human mind will never solve.”
Human beings have always studied prime numbers and its divisors. The truth is that we should have looked at their multiples.
I have done that and here is the orbital sequence of the prime numbers : The SOP puts order in the set of prime numbers.
The Orbital Prime Number Sequence » SOP» in which all the prime numbers are stored.
It is a succession like wagons of a train, whose machine is in infinity.
Each wagon there is a number, label, which is the product of all the prime numbers that are inside each wagon, there are the prime numbers that belong to it and that we will call brother primes.

Index

Introduction

1.- Nomenclature. The giant numbers.

2.- Definition and algorithm In, calculation of the terms of the SOP.

2.1.-Formulation and properties of giant numbers. The Enlargement.

3.- The first 100 terms of the SOP.

4.- «TodoUnos» Theorem. «TodoNueves» Algo- rithm. Algorithm of the «Anterior»

5.- Calculation of the period of a number. The number of digits in that period. Various proper- ties.

5.1.- Algorithm of the «Anterior»(2) . «Primo superior» Theorem . «Número Compuesto» Theorem.

6.- Congruences and «Números repetidos», Theo- rem (TNR) and its corollaries. «Continuidad» The- orem.

7.-Cyclic primes. List of the In to which they be- long. Looking for primes

8.- The SOP. Test of prime number, and order. Subsequences of the SOP.