On the Binary Goldbach Conjecture: Analysis and Alternate Formulations Using Projection, Optimization, Hybrid Factorization, Prime Symmetry and Analytic Approximation

An analysis, based on different mathematical approaches, of the binary Goldbach conjecture −which states that every even integer s≥6 is the sum of two odd primes, called Goldbach primes− is presented. Each approach leads to a different reformulation of this conjecture, thus contributing unique insights into the structure, properties and distribution of prime numbers. The above-mentioned reformulations are based on the following distinct, interrelated and complementary approaches: projection, optimization, hybrid prime factorization, prime symmetry and analytic approximation. Additionally, it is shown that prime factorization is an optimal projection operation on the set of integers; that Goldbach pairs correspond to solutions of an optimization problem; that hybrid prime factorization can be used to generate Goldbach primes; that prime symmetry, a powerful property of Goldbach primes, can be used to validate the binary Goldbach conjecture in short intervals, and to determine the rules that govern the “algebraic evolution” of Goldbach pairs, as the value of s increases; and that analytic approximation, using translational and rotational shifts of smooth functions, leads to a useful approximation of a primality test function and the prime counting function π(s). The paper’s findings support the broader hypothesis that prime numbers, by virtue of their optimality in representing, additively and multiplicatively, any measurable quantity in the universe, supported by the Fundamental Theorem of Arithmetic and the binary Goldbach conjecture, may be a viable alternative to the exclusive use of binary logic, as a means of achieving additional computational efficiencies of scale in the future.