In a historic breakthrough The Fundamental Four equations create and define Potential Primes (Pp), Elimination Values (EV), and Prime numbers (Pp'). This offers a significant addition to understanding and a rigorous mathematical proof of the origins and nature of prime numbers. 
 
Summary of The Breakthrough:
 
The Fundamental Four equations that define all integers from 2 to infinity.
 
Twin Prime Conjecture Proof:
Using the Pp, EV, and Pp' framework, a rigorous proof of the Twin Prime Conjecture is shown. This resolves a problem that has remained open since Euclid's time, proving that there are indeed infinitely many pairs of primes that differ by 2.
 
Strong Goldbach Conjecture Proof:
A comprehensive and rigorous proof of the Strong Goldbach Conjecture is provided, showing that every even integer greater than 2 can be expressed as the sum of two primes. This settles a question that has puzzled mathematicians since 1742.
 
Weak Goldbach Conjecture Proof:
Similarly, this framework provides a stringent proof of the Weak Goldbach Conjecture, demonstrating that every odd integer greater than 5 can be expressed as the sum of three primes.