An Analysis of Approaches to Goldbach's and De Polignac's Conjectures and Their Interconnections

Goldbach's Conjecture and de Polignac's Conjecture remain two of the most persistent unsolved problems in the additive properties of primes and have been so for nearly two centuries. This paper attempts a thorough exercise on major analytical methods currently available for adjudging these conjectures. Analytically, this paper discusses two main theorems that have been used to address Goldbach's Conjecture: that of Chen Jingrun and the Hardy-Littlewood Circle Method. Similarly, the discussion provides the Goldston-Pintz-Yıldırım sieving, which relates to bounded prime gaps, and critically surveys a specific purported proof of de Polignac's Conjecture from 2014. It also reports the results of a computational verification done to prove the conjecture for evens up to 10,000, together with an associated property. This turns out to be with defects in the earlier purported proof. The paper ends by reiterating the closeness of the Conjectures-in demonstrating their deep entanglement with the prime number distribution function-as well as the difficulty inherent in connecting two apparently disparate behaviors of the same object, even though both are tied to their additive disposition.