Advancements in Prime Number Study and the Non-Existence of Odd Perfect Numbers

This paper presents original contributions to two longstanding areas of number theory: the non-existence of odd perfect numbers, and the structure and distribution of prime numbers with emphasis on prime gaps and implications for the Riemann Hypothesis. The first section of the paper explores the hypothetical existence of odd perfect numbers. We begin by assuming the existence of such a number and then contradict the assumption by a recursive quadratic equations involving its divisors. In the second section, we turn our attention to the distribution of primes. We propose a structural classification of twin primes based on ending digits, and prove that there are infinitely many twin primes by contradiction. We further propose a new upper bound for prime gaps, specifically the inequality $P_{n+1} - P_n < 2\sqrt{P_n}$. This is proved using explicit bounds for the prime-counting function \( \pi(x) \), particularly the inequalities established by Rosser and Schoenfeld. Finally, we address the Riemann Hypothesis by showing that for all sufficiently large \( x \), the interval $\left(x - \frac{4\sqrt{x} \log x}{\pi},\ x\right]$ must contain at least one prime. This result, originally connected to the Riemann Hypothesis by Adrian Dudek, is approached here via contradiction and comparison to our earlier prime gap bound. Altogether, this paper contributes new proofs, bounds, and structural insights into some of the deepest unsolved problems in number theory.