In this paper several methods are examined for proving the Goldbach conjecture. At the preliminary analysis stage a Diophantine equation solution method is proposed for Goldbach partition of a Goldbach number. The proof method proposed however is found to be incomplete since it does not have mechanisms for dealing with the prime gap problem. On the further analysis section some graphical and linear analytical methods are proposed for Goldbach partition as an extension of the solution of proposed quadratic equation. The Riemann hypothesis is examined in light of some findings on Goldbach conjecture. A proof is then proposed for the Riemann hypothesis. The proof results are used to attempt to prove Goldbach conjecture but without success. A justification for proof by induction method is proposed. A theorem 1 is proposed by an attempt is made to prove the conjecture by induction. To reinforce the proof by induction, a Samuel –Goldbach theorem is proved in which it is shown that any even number greater than six is the sum of four prime numbers. The theorem is then reduced to Goldbach strong and weak conjectures. Goldbach weak conjecture (proved) is also reduced to the strong conjecture. A proof method is thus proposed by which the weak conjecture is reduced to the strong. The proof method however is not completely satisfactory because it does not provide an analytical solution of the prime gap problem. Proof method however gave lead to the importance of even numbers in Goldbach partition. A proof method of proving the Goldbach conjecture is discussed by which each odd prime number is connected to a specific even number. Through this connection a family of curves with even number points for Goldbach partition of a Goldbach number is proposed. The families of curves containing these special even coordinate points help overcome the prime gap problem in Goldbach partition. It is found that each Goldbach number has at least one pair of these special even numbers to enable Goldbach partition. A special identity then used to come up with a special quadratic function for Goldbach partition. The function has at least one point with an x coordinate representing gap between primes of the Goldbach partition any a y coordinate that is a product of the same primes. Thus Goldbach conjecture is fully proved and the prime gap problem of the partition solved.

Keywords: Goldbach integers; Primes; Riemann Hypothesis; Goldbach strong Conjecture; Goldbach weak conjecture; Twin prime conjecture,