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PREFACE

THE SEQUENCE OF THE PRIMES IS SOMETHING OF A MYSTERY in number theory, if not in all of mathematics. Hardly any topic has ever fascinated mathematicians more.

Leonhard Euler once said, “Mathematicians have tried in vain, to this day, to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate”.¹

One key question has always been: Does there exist a single, overarching organisational principle that accounts for why any prime follows another prime as the next largest prime in the prime sequence?

The American mathematician Don Bernard Zagier wrote, “Despite their simple definition and role as the building blocks of the natural numbers, the prime numbers grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout”.²

Prime numbers have fascinated matematicians and scientists for centuries, and their history is closely related to the very history of Mathematics. Interest in prime numbers has consistently found new motivation because they have appeared in different contexts, ranging from Cryptology to Biology, and Quantum Chaos. However, despite the huge advances in number theory, many properties of prime numbers remain unknown, and they appear to us as a weird random collection of numbers without much structure. In the last few years, some numerical investigations related to the statistical properties of the prime number sequence have revealed that, apparently, some regularity actually exists in the differences and increments (the differences of the differences) of consecutive prime numbers.³ The English mathematician G. H. Hardy wrote, “The ‘average’ distribution of primes is very regular; its density shows a steady but slow decrease. On the other hand the distribution of the primes in detail is extremely irregular.” A theory established by mathematicians over centuries of study.

A primary aim of this book is to show that such an overarching and comprehensive principles do exist and govern the entire prime sequence. To unlock these secrets of harmony, for the enthusiastic reader, it is not necessary to dive too deep into mathematics, but to unveil structures of organisation from different points of view.

The Secret Harmony of Primes was written in 2012-2015 and represents a substantial part of my work in prime number theory throughout the last decades, woven into a comprehensible storyline.

Sam Vaseghi, Stockholm

June 2015