The binary Goldbach conjecture asserts that every even integer greater than 2 can be expressed as the sum of two primes. In addition, the ternary Goldbach conjecture states that every odd integer greater than 5 is the sum of three primes. Both conjectures originated in an exchange of letters between Leonhard Euler and Christian Goldbach in 1742. We claim that both Goldbach problems are valid and have multiple partitions for every integer greater than 12. We approach the binary Goldbach problem by the Second Principle of Mathematical Induction. Based on the inductive hypothesis, we prove that every even integer within a finite interval can be expressed as the difference between two primes. This fact allows us to tackle the inductive step advantageously. Using the binary and ternary Goldbach problems, we prove that every even integer can be written infinitely many ways as the difference between two primes, where the twin-prime problem is a particular case.
MSC Classification: 11P32 , 11N05