A New Approach to Prime Numbers and Why Golbach’s Conjecture is True

Abstract: A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. The crucial importance of prime numbers to number theory and mathematics in general stems from the fundamental theorem of arithmetic, which states that every integer larger than 1 can be written as a product of one or more primes in a way that is unique except for the order of the prime factors. Primes can thus be considered the “basic building blocks”, the atoms, of the natural numbers. There are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no known simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, the statistical behavior of primes in the large, can be modelled. The first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability that a given, randomly chosen number n is prime is inversely proportional to its number of digits, or to the logarithm of n. The way to build the sequence of prime numbers uses sieves, an algorithm yielding all primes up to a given limit, using only trial division method which consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. This paper introduces a new way to approach prime numbers, called the DNA-prime structure because of its intertwined nature, and a new process to create the sequence of primes without direct division or multiplication, which will allow us to introduce a new primality test, and a new factorization algorithm. As a consequence of the DNA-prime structure, we will be able to provide a potential proof of Golbach’s conjecture.