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
"To some extent the beauty of number theory seems to be related to the contradiction between the simplicity of the integers and the complicated structure of the primes, their building blocks. This has always attracted people." 4
Natural numbers are those numbers that have two main purposes: counting and ordering. In mathematical terms, they are cardinal and ordinal.
There is no agreement on whether the number ‘zero’ is a natural number too. This is why we still use two different mathematical notations for the set of all natural numbers;
and
Figure 1.1: The sequence of natural numbers as a linear plot.
The set of natural numbers is infinite but countable. Generally, and including zero, one can apply a recursive addition to all natural numbers, beginning with
that reads “
Through recursive addition every natural number is tied with an additive property in relation to other natural numbers:
This brings us closer to what is called a total order of the natural numbers. We can have
But natural numbers are not only in total order, they are also well ordered: every non-empty set of natural numbers has a least element. There exists, at any time, a rank among the sets that can be expressed by an ordinal number
Figure 1.2: Mayan numerals.
Given that with
In this way, every natural number is tied to a multiplicative property in relation to other natural numbers:
These properties of addition and multiplication mean that natural numbers emerge as an instance of a commutative semiring. They cannot be called a ring because
Although there is a defined procedure of division with remainder, it is not possible, in a generalised way, to divide a natural number
In mathematical terms, the procedure of division with remainder can be expressed for any two numbers
