What question did this study set out to answer?

The research aims to establish a formal proof for both the Riemann Hypothesis and the Twin Prime Conjecture.

June 9, 2026Open Access

The Prime Questions

Key Points

The research aims to establish a formal proof for both the Riemann Hypothesis and the Twin Prime Conjecture.
Defined a continuous geometric manifold M bounded by the cumulative sum of prime numbers Dn.
Mapped the discrete prime distribution to a complex vector space.
Proved the existence of infinite prime pairs and the implications of non-trivial zeros of the Riemann Zeta function.
Proved that structural continuity of manifold M ensures infinite prime pairs differing by 2.
Established that any non-trivial zero off the critical line of the Riemann Zeta function causes topological collapse, confirming the Riemann Hypothesis.

Abstract

This paper presents a formal proof of the Riemann Hypothesis and the Twin Prime Conjecture. We define a continuous geometric manifold M strictly bounded by the cumulative sum of prime numbers Dn. By mapping the discrete prime distribution to a complex vector space, we prove that the structural continuity of M forces the existence of infinite prime pairs differing by 2. Furthermore, we provide a definitive proof that any non-trivial zero of the Riemann Zeta function lying off the critical line ℜ(s) = 1/2 causes a topological collapse of the manifold, thereby proving the Riemann Hypothesis by contradiction.

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