Principles of prime numbers - Part IV - Twin prime & Polignac conjectures – Conservation of gaps

Author(s): James M McCanney*

The prior papers “Principles of Prime Numbers - Part I - New Definition of Prime Numbers with ModNt Number System & Induction”, “Principles of Prime Numbers - Part II - Anatomy of Spp_n & Rpp_n Tables with Conversion from mod10 to modNt Number System” and “Principles of Prime Numbers - Part III - Spp_n & Rpp_n Table Parameters with Advanced Topics and Twin Prime Conjecture” must be read to put this paper into perspective. Those papers are a summary of three books “Calculate Primes” (2007), “Principles of Prime Numbers – Volume I” (2010) and “Breaking RSA Codes” (2014). Another peer reviewed paper can be included in the list of prior papers “New Definition of Prime Numbers with Spp_n Tables and Proofs by Induction”.

In the prior books and papers, the concept of direct calculation of prime numbers was presented using the McCanney Generator Function and a sole boundary condition. It showed that the prime numbers constitute a complete number system modNt based on sequential prime products Sppn with visual representation in the form of “Spp_n and Rpp_n tables”, creating the basis for proofs by induction for the Twin Prime Conjecture and other problems. The groups of prime numbers form algebraic groups. Groups of prime numbers generate larger groups beginning with Peano’s Postulates. The new mathematical system has properties of closure, completeness, reciprocity, symmetry and generates the wave equations “comb_n ” that predict all future primes to infinity. Each prime number or series of prime numbers has a unique ancestry. This is demonstrated in the digits of the modNt number system. Likewise, each prime or group of primes has an infinite number of descendants. Composite numbers that are relatively prime to the values of Spp_n are important in the development of real primes, twin primes and larger groupings of real primes. Without these relative primes, the process would not generate all real primes and would not be complete. Prime numbers belong to families with ancestors, descendants as well as siblings.

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