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 matheÂŹmaticians more. 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". 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? A primary aim of this book is to show that such overÂŹarching 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 collection of the author's work in prime number theory throughout the last decades' woven into a comprehensible storyline.
"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
is not a number
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 âzeroâ is a natural number too. This is why we still use two different mathematical notations for the set of all natural numbers;
that excludes zero:
and
that includes zero:
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
and succeeding with:
that reads â
plus a successor
equals the successor
â.
Through recursive addition every natural number is tied with an additive property in relation to other natural numbers:
is a commutative monoid with identity element
, the free monoid with one generator. Monoids are algebraic structures with a single associative binary operation and an identity element; they are categories with a single object.
This brings us closer to what is called a total order of the natural numbers. We can have
if, and only if, there exists another natural number
with:
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
addition has been defined, a multiplication
can also be defined, beginning with
and succeeding with:
In this way, every natural number is tied to a multiplicative property in relation to other natural numbers:
is a so-called free commutative monoid with identity element
.
These properties of addition and multiplication mean that natural numbers emerge as an instance of acommutative semiring. They cannot be called a ring because
is not closed under subtraction and lacks an additive inverse.
Although there is a defined procedure of division with remain...
