Reviews

Dr. Scott A. Vanstone: “Do you have any idea what this means?”

 

• PhD (Mathematics) — University of Waterloo.

• Distinguished Professor Emeritus of Mathematics and Computer Science, University of Waterloo; Taught for 35 years; 125 Patents; 150 Research Papers.

• Authored several books including “The Handbook of Applied Cryptography” and “A Guide to Elliptic Curve Cryptography”

• Fellow of the Royal Society of Canada, Academy of Sciences — 1998.

• Premier's Award for Lifetime Achievement in Innovation — 2009

• Fellow of the International Association of Cryptologic Research (IACR) — 2011.

 

"I haven't slept for four nights. It is magic!" - Roydon Fraser. - Professor, PEng, FEC, Teaching Chair – University of Waterloo.

 

”This will shake the foundations of computer science!"  - Aaditeshwar Seth. PhD. Computer Science.

 

"I have read your proof, it's really impressive and straightforward to read. I'm really impressed with the results of this work. My first reaction, when I understood it was: Wow!". Primes play a huge role in modern cryptography, main cryptographic algorithms such as RSA, Diffie-Hellman, DSA are based on primes. That means that these results can create a revolution in cryptography!  - Tatyana Khitrova. Computer Scientist and Cryptography expert.

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"What is found in this book is far different. It extends far beyond 6k ± 1 and the current understanding of prime numbers in Number Theory. Indeed, it provides rigorous proofs of the equations defining the prime number sequence, the Twin Prime Conjecture, and the Goldbach Conjectures."

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Highlights of The Fundamental Four

The world of mathematics has been forever changed by a series of remarkable breakthroughs centered around four seemingly simple equations, which are called the "Fundamental Four." These equations have not only solved long-standing problems in number theory but have also opened up new avenues for mathematical exploration and practical applications.

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The significance of these accomplishments cannot be overstated. They represent a fundamental shift in our understanding of prime numbers and their properties, potentially revolutionizing fields such as cryptography and opening up new avenues for mathematical research.

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Summary of The Breakthroughs:

 

• The Fundamental Four:

         The Fundamental Four has been proven to define all positive integers from 2 to             infinity. These equations are very much like the DNA of integers and prime                       numbers.​

• Equations of Prime Numbers Proven:

  We have established a revolutionary framework for determining and precisely characterizing prime numbers using the Fundamental Four equations to derive Potential Primes (Pp), Elimination Values (EV), and Prime Candidates (Pp'). This framework provides a complete characterization of the prime number sequence.

• Twin Prime Conjecture Proof:

  Using the Pp, EV, and Pp' framework, a rigorous proof of the Twin Prime Conjecture has been 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:

  This has provided a comprehensive and rigorous proof of the Strong Goldbach Conjecture, 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.

 

The Fundamental Four not only settles centuries-old conjectures but also provides a new lens through which to view the fundamental nature of numbers. The implications of these discoveries extend far beyond pure mathematics:

 

• New approaches to cryptography and computer science

• Potential breakthroughs in analytic and algebraic number theory

• More efficient algorithms for prime generation and primality testing

• Novel educational tools for teaching number theory

 

These breakthroughs not only resolve specific problems but also provide a new paradigm for approaching fundamental questions about numbers.

 

James P. Moore

September 24, 2024

Waterloo, Ontario. Canada