This paper addresses two major topics in number theory: the non-existence of odd perfect numbers and the distribution of prime numbers, particularly concerning about prime gaps In the first part, I provide a rigorous contradiction-based proof of the nonexistence of odd perfect numbers by examining recursive divisor equations of the form $q^2 - 2mq + 2m + 1 + qL + qV + qA + qB + \dots = 0.$. I analyze the parity of $q$ and the discriminant to show that such structures cannot exist under odd constraints.Through the analysis of recursive equations and the properties of divisors of hypothetical odd perfect numbers, I demonstrate that no odd perfect number can satisfy the necessary conditions, thereby supporting the long-standing conjecture on their non-existence In the second part, I explore twin prime behavior using digit pattern analysis and refine the study of prime gaps by establishing a new upper bound for the difference between consecutive primes (i.e)$ P_{n+1} - P_n < 2\sqrt{P_n}$ via known results shown by J Barkley Rosser and Lowell Schoenfeld. The findings presented in this paper offer significant advances in the study of perfect numbers and prime number theory, providing new insights into both classical and modern problems in number theory. These results open the door for further exploration and deepen our understanding of these fundamental mathematical concepts.