The Mathematics behind Cook's Riemann Hypothesis Proof & More

The Fundamental Theorem of Arithmetic. "Every natural number greater than 1 can be written as a unique product of prime numbers." And thus, all else follows. The very notion of numbers is built upon the multiplication of primes. This theorem so facinated Jeff Cook that he became obsessed with the depths of pure mathematics.

But first...

There are two primary stages of mathematical research that one generally comes across, and really need to get a handle on before any of Cook's research is dealt with: 1) Conjecture & Hypothesis and 2) Theorems and Proofs. The purpose of grouping conjectures with hypotheses and theorems with proofs is simple: often one or the other is expendable depending upon where you are led in your research. While it is not possible to write a proof without theorems as your foothold, it is possible to write proofs around and about a hypothesis or conjecture without having first proven the specific hypothesis or conjecture in question. And while you can certainly have a hypothesis without ever needing to have a conjecture, as one or more aspects of the hypothesis may rely on previous proofs without ever having to assume a thing. There is an order to such processes, but first the basics should be defined.

Conjecture: a simple guess with little or no mathematical basis.

Hypothesis: a statement of what you think and why.

Theorem: a mathematic fact that has already been proven.

Proof: a structure built upon known theorems that can show that a conjecture or hypothesis is true.

In the following pages, only pure mathematics will be discussed with little or no relation to other sciences, except perhaps to illustrate a particular point from time to time for simplicity sake. And most of what follows are the developments in pure mathematics that Jeff Cook has made over the years, which will include at least the following: prime numbers, the Riemann Hypothesis and Mr. Cook's proof that it is true, Neutronics, Anarithms and Opponents and most definitely (pun intended) his Definitive Theorem.