This research paper presented the development of a formula for odd-prime numbers and showed its powerful application in proving longstanding conjectures in number theory. The method involved the development of a formula for odd-composite numbers and investigation of its properties to establish a connection with primes. Key results of this study include the derivation of an odd-prime formula which directly helped provide complete proofs to longstanding problems in prime number theory: the Goldbach, Polignac, and Twin Prime Conjectures. It offers a significant insight that made a big contribution in providing solutions to these problems which remained unsolved for centuries.