New Algebraic Proofs on the Correctness of Collatz Conjecture basing on New Unified Formulas along with Computational Results which Provide Insights on Prime Numbers

This paper presents new mathematical proofs on the correctness of the Collatz conjecture based on new unified formulas re-expressing this conjecture in a scalable algebraic form. We are proposing new precise formulas that enable expressing all natural numbers in the Collatz conjecture according to a unified methodology that allows scaling calculations on infinite numbers, and then we use these unified formulas to prove that all natural numbers are converging to the number one when we perform Collatz operations on them. In addition, we are presenting computational codes based on these algebraic formulas to provide computational results that guide the development of proofs. As a result, this paper is presenting twenty seven new theorems along detailed proofs, where the first seven theorems are proposing new unified formulas to re-express the values of numbers according to unified algebraic forms, whereas the other theorems are using these new formulas to demonstrate that all natural numbers are converging to the number “1” when we forward Collatz operations on them; whereas demonstrating that there is no divergency over these operations. We also use these formulas to demonstrate that all natural numbers create loops in infinity that may cause Collatz calculations to circulate in a ring of numbers. Furthermore, we demonstrate that the only ring of natural numbers that creates a loop for the Collatz conjecture is where we go from “1” to “4”, then calculations converge back to the number one. All theorems presented in this paper are developed according to an engineering methodology based on structuring unified formulas to re-express Collatz calculations along step-by-step proofs, which allow us to develop a breakthrough demonstrating the correctness of the Collatz conjecture, while providing new insights into the characteristics of prime numbers and their distribution.