Abstract

In this research a necessary and sufficient condition for the proof of the Binary Goldbach conjecture is established. It is established that the square of all natural numbers greater or equal to 2 have an additive partition equal to the sum of the square of a natural number greater or equal to zero and a Goldbach partition semiprime. All Goldbach partition semiprimes are odd except 4. This finding is in itself proof that all composite even numbers have at least one Goldbach partition. The result of the proof of the Binary Goldbach conjecture is used to prove the Andrica and Legendre conjectures. The Riemann hypothesis is examined and sources of non trivial zeroes outside the critical strip are discussed. An example example of a non-trivial zero outside the critical strip is given. An exact generalization of gaps between consecutive primes is brought to light to enable further insights about twin primes and small gap primes in general. A function for counting the number of Goldbach partition is derived. Furthermore a function for calculating the average prime gap is also derived.

Keywords: Proof of binary Goldbach conjecture, Proof of Andrica conjecture, Proof of legendre conjecture, Goldbach partition semiprime, Disproof of Riemann hypothesis, Proof of twin prime conjecture, Goldbach partition counting function, Average gap between primes % create a new 1st level heading