INTRODUCTION

In ordinary and superior mathematics many advances have been conducted in the domain of measurement and order, since the discovery and the "demonstration of existence", by the greek mathematician Euclid (328-265 BC) in Book IX of the Elements, of the elegant theorem which states that "prime numbers are an infinite series".

New concepts, methods and techniques let solve ancient and complex problems related to the prime numbers, their formation and the criteria to decide if any number is or is not a prime. But, the reciprocal is impossible, i.e: nobody has yet shown a simple arithmetic formula to generate prime numbers, although the modern theory of numbers has appealed to radical means (see DECODING THE SECRETS OF THE LAW OF PRIMES IN NUMBER THEORY, 10/XI/01, by the same author of this research).

THE STRUCTURE OF ARITHMETIC

The advances made by the greek mathematicians in relation to the decimal system of numeration let them define the concept of a prime: "is the number which is divisible by itself and 1". Eratosthenes (284-192 bC) built his famous sieve to, intuitively and later formally, systematize a procedure which lets find all prime numbers, with the assumption of devoting hard work and enough time, under a simple reasoning and yielding an easy expression.

With the formula n+1 all positive integers are listed in their natural order: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,…: then the multiples of the successive numbers are eliminated by the proper divisors, except the two trivial divisors "by itself and 1", hence obtaining the series of prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97,…with this siege.

METHODOLOGY OF THE PRESENT RESEARCH

A first conjecture to show states that the prime series becomes scarce as it advances in its development with larger gaps between successive primes, without letting foresee any regularity in the decimal numerical field of study.

It is a laborious process. It has been estimated that over 300 hours of hard work are needed to "sieve" all prime numbers comprised between 1 and 1.000.000, although, observing that they get scarce with larger and larger gaps which, naturally, leads to the question or hypothesis: Is there a maximum prime? or, Do twin primes exist no more somewhere along the series? Indeed, a real challenge for professionals and amateurs.