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   Introduction     

The strong Goldbach conjecture and the weak Goldbach conjecture are covered. For comparison, the two conjectures are also covered side-by-side. The strong Goldbach conjecture states that every even integer greater than 4 can be expressed as the sum of two odd primes. The weak Goldbach conjecture states that every odd integer greater than 7 can be expressed as the sum of three odd primes. The approach in the coverage of the weak Goldbach conjecture is similar to the approach used in proving the strong conjecture. However, two approaches for producing Goldbach partitions for the weak conjecture are covered. In the first approach, one applies the principles used in finding partitions for the strong conjecture. Beginning with the partition equation, 9 = 3 + 3 + 3, and applying the addition of a 2 to both sides of this equation, and subsequent equations, one obtained Goldbach partitions for 26 consecutive odd integers. In the second approach, one produces the partitions from the partitions of the strong conjecture by adding a 3 to both sides of a strong conjecture partition equation. For the strong conjecture, one will begin with  the partition equation 6 = 3 + 3, and apply the addition of 2 to both sides of the equation to produce the partition for the next even number, 8. From the partition equation, 8 = 5 + 3, one will repeat the 2-addition process to obtain the partition for the next even integer, 10. From the partition for 10, the process can continue indefinitely. This repetitive process was compared to the repetitive process in compound interest calculations. It is shown that given an equation for a Goldbach partition, one can produce a Goldbach partition for any even integer greater than 4 as well as produce a partition for any odd integer greater than 7. A consequent generalized procedure also produced Goldbach partitions for non-consecutive even and non-consecutive odd integers. In addition to directly producing partitions of the strong conjecture, one can also produce partitions of the strong conjecture from the partitions of the weak conjecture and vice versa. Formulas derived for the Goldbach partitions show that every even integer greater than 4 can be written as the sum of two odd prime integers; and also that every odd integer greater than 7 can be written as the sum of three odd prime integers. Importantly, in addition to showing that the Goldbach conjectures are true, shown also is how to produce Goldbach partitions. Three proofs of the Goldbach Conjectures are presented. A prize of one million dollars is being offered by the publishing house, Faber and Faber, for a proof of Goldbach Conjecture.

                                                            Proofs of the Goldbach Conjecture

Proof #1: Goldbach Conjecture Proved Remarkably

Proof #2: Goldbach Weak Conjecture Proof

Proof #3: Strong & Weak Goldbach Conjectures Proved Side-by-Side

 

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