Goldbach’s Conjecture

We discuss the Goldbach Conjecture, a famously unsolved problem in number theory. One of its iterations states that every positive odd number greater than 7 can be written as the sum of three odd primes. In particular, we consider Christian Goldbach and Leonard Euler’s correspondence and their early discussions of the Goldbach Conjecture. We also state derivations of the problem, including the ternary, strong, and weak versions of the Goldbach conjecture. We also review the ideas behind the efforts to prove the conjecture, specifically some brute force attempts, Ivan Matveevich Vinogradov’s theorem, and Harald Andrés Helfgott’s recent proof on the weak Goldbach conjecture. Finally, we examine the Goldbach Conjecture’s significance and role in number theory and mathematics as a whole by relating it to other famously unsolved problems, such as the Riemann Hypothesis. We also explore how the conjecture can be used in other fields of mathematics, such as in cryptography and in group theory.