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This paper introduces a new method to find the next prime number after agiven prime ${P}$. The proposed method is used to derive a system ofinequalities, that serve as constraints which should be satisfied by all primeswhose successor is a twin prime. Twin primes are primes having a prime gap of${2}$. The pairs ${(5,7),(11,13),(41,43)}$, etcetera are all twin primes. Thispaper envisions that if the proposed system of inequalities can be proven tohave infinite solutions, the Twin Prime Conjecture will evidently be proventrue. The paper also derives a novel upper bound on the prime gap, ${G_{i}}$between ${P_{i} \; and \; P_{i+1}}$, as a function of ${P_{i}}$.

Towards Proving the Twin Prime Conjecture using a Novel Method For  Finding the Next Prime Number ${P_{N+1}}$ after a Given Prime Number  ${P_{N}}$ and a Refinement on the Maximal Bounded Prime Gap ${G_{i}}$