The Orbit Theory of Natural Numbers Volume II: Applications to Number Theory

At times, when a mathematical proposition is hard to prove, it is because the appropriate concepts and tools have not been formulated. Orbit Theory presents a new set of tools for examining some of these unproven conjectures. Volume I presents Fundamentals of Orbit Theory that looks under the hood of the Natural Number line revealing the engine underneath. A major component of this engine is the Orbit Table, a collection of values distinct for each Natural Number. Put into a table, Orbit values of one number relate in various ways to Orbit values of other numbers. These are shown to create “structures” that can repeat throughout the Orbit Table. Volume I introduces elementary structures such as Verticals, Rays, Triangles, and Diamonds. This volume goes much deeper in examining more profound structures and how they relate to some of the unproven conjectures of Number Theory. Volume II provides insight into Goldbach’s Conjecture, that every even number is the sum of two prime numbers, as well as Maillet’s Conjecture, that every even number is the difference of two prime numbers. It explores patterns of Orbits for Perfect Squares, Triangular numbers, BiPrimes, Twin Primes, Cousin Primes, Sexy Primes, and Prime Triplets. Orbit Theory shows how the Orbit Table unlocks some of the secrets of Legendre’s Conjecture, Brocard’s Conjecture, Oppermann’s Conjecture, Grimm’s Conjecture, and other famous problems in Number Theory. The Orbit structure of Sophie Germain Primes hold a special beauty in that a “fingerprint” can be defined that is found in every Sophie Germain Orbit Structure. Two fundamental properties of the Orbit Table come to light in this investigation, the Repetitive Segment Property, where the same sets of Orbit values cross all Natural Numbers regardless of how big the numbers become; and the Three Rays Property, consisting of the Vertical Ray, the Left Ray, and the Right Ray for a number, all having the same Orbit values across each row. These two properties are demonstrated to hold for functions defining even numbers, odd numbers, powers of numbers, and polynomials of numbers, regardless of how many columns are removed from the Orbit Table. Orbit Theory is in its infancy. It is the hope of the author that these two volumes spark enthusiasm within the math community by professional mathematicians, professors and graduate students of mathematics, as well as the amateurs who are simply astounded by the beauty of the Orbit Table.