Since ancient times conisseurs of number theory are fascinated by the prime-composite classification of natural numbers, but the law governing their distribution could t be discovered. This problem was also highlighted by CF Gauss by stating: the dignity of the science itself seems to require that every possible means be explored for the solution of the problem so elegant and so celebrated. This mograph provides the solution to this problem by studying primes via composite numbers and, interestingly, it is simple eugh to be understood even by under-graduates. The formula so obtained has found several applications, while leaving further scope for the specialists. Thus, it meets the requirement of amateurs and specialists alike. APPLICATIONS: An effective algorithm for separating primes from composite numbers along with their prime factors developed; a new formula for number of primes up to any given integer sans approximation and hypothesis derived; Dirichlet's prime number theorem modified; three prime conjectures proved; five new arithmetic functions introduced; some holistic probability-theoretic invative ideas developed by going in to the genesis of the problems.