The Secret Harmony of Primes

CHAPTER 1

Fundamentals

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 ‘zero’ 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 plus a successor equals the successor .

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 a commutative 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 remainder, it is not possible, in a generalised way, to divide a natural number by another natural number , and expect to get a natural number as the