Page ix - Prologue

Prologue

“If you have an apple and I have an apple and we exchange these apples then you and I will still each have one apple. But if you have an idea and I have an idea and we exchange these ideas, then each of us will have two ideas.”

George Bernard Shaw

My fascination with prime numbers started before I can remember. I was always sure there must be patterns governing their distribution and also I felt that the understanding of these patterns was in my grasp.

I knew I was in good company. Over the millennia all the great mathematicians have tried to find these patterns, and most of them felt that there must be such patterns. It was Don Zaghier in his well known paper "The first 50 Million Prime Numbers" that said it best:

"There are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts.

The first is that, despite their simple definition and role as the building blocks of the natural numbers, the prime numbers belong to the most arbitrary and ornery objects studied by mathematicians: they grow like weeds among the natural numbers, seeming to obey no other law then that of chance, and nobody can predict where the next one will sprout.

The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behaviour, and that they obey these laws with the almost military precession."

By taking on such a well known problem I proposed to myself to find first the best approaches that could lead me to a solution. I was kind of aware of the difficulty and complexity of the problem, but the real implication and the depth of the solution only revealed to me after long and continuing meditation and profound immersion of all my thoughts into the problem during many years.

Looking back in history to men with the most relevant contribution to the development of mathematics and science, few methods and principles, applied by them, stand out. Following them, these are the main guidance principles I followed during my investigations.

1. Take a geometrical approach. It is always better to use a geometrical construction when trying to explain a complicated phenomena or trying to understand a complicated pattern. There are at least two reasons that make this a better approach. First, we experience life in a multidimensional space that exposes us to a richer variety of patterns and phenomena than those available in a linear type, formal approach. Our brains have developed to cope with the complexity of the multidimensional space. I will argue that there is no better method to communicate complex ideas then geometry. Second, the space as shown by Riemann is just one example of a multiple extended