Prime Number Research

Introduction

The Greek philosopher and mathematician Pythagoras (500-300 B.C.) and the Pythagoreans were very interested in numbers; they were fascinated by them and saw them everywhere, to the point of saying that everything is a number, and among these numbers, they gave immense importance to prime numbers.

In arithmetic, which is one of the branches of mathematics, number theory deals with prime numbers, and generally with the properties of natural integers, relative integers, and rational numbers (in the form of fractions). It also deals with the properties of the operations of addition, subtraction, multiplication, and division on these numbers.

The prime number is defined as: "If the two positive whole divisors of a natural integer are only 1 and itself, then that natural integer is a prime number." Any other natural integer greater than 1 is said to be a composite.

Despite its simple definition, the prime number is the simplest and most complicated mathematical object at the same time. For mathematicians, and for centuries, it has been one of the most intriguing mysteries. Thanks to a very intelligent and logical reasoning, the famous Greek mathematician Euclid (300 A.D.) managed to prove successfully that prime numbers are infinite.

Scientific Utilities

The scientific content of this work contributes to the evolution of several fields:

• In arithmetic: Prime numbers are essential for the simplification of rational fractions.
• In computer science: Gigantic prime numbers that are difficult to factor are used to test the reliability and speed of large calculators.
• In cryptography: Prime numbers are crucial for secure internet banking transactions and the transmission of sensitive information, protecting private data and securing highly confidential messages.

Abstract of Research

This study about prime numbers has achieved several results, including:

• A crucial method to obtain any given quantity of prime numbers using two sets:
  • The set F and its elements (6a - 1)
  • The set G and its elements (6a + 1)

  The numbers (6a - 1) and (6a + 1) are:

  • Odd numbers superior to 3; not multiples of 3; composites.
  • Odd numbers superior to 3; not multiples of 3; primes forming a set noted ℙ.
• Determination of the set ℙ of prime numbers of the form (6(a) - 1) and (6(a) + 1) using eight remarkable two-variable formulas that yield:
  • All composite numbers in F of the form (6(a) - 1) to deduce the prime (6(a) - 1).
  • All composite numbers in G of the form (6(a) + 1) to deduce the prime (6(a) + 1).
• Primality test, decomposition, and divisors of any composite integer using new techniques based on frames and formulas.
• Formulas to determine the set of prime numbers ending in the digits 1, 3, 7, and 9.

Key Publications

• "FIND OUT ALL THE PRIME NUMBERS BY FORMULAS OF NON PRIMES" (2021, 165 Pages)
• "THE DISCOVERY OF ALL THE PRIME NUMBERS BY PRECIOUS FORMULAS OF THE NON PRIMES" (2022, 337 Pages) - Expanded version of the first work.