Year: 2026
Type: preprint
Abstract: This paper presents a simplified proof mechanism for the Twin Prime Conjecture using a proof by contradiction strategy termed the "Best Way." Assuming a finite bounds model where a final terminal pair \((p_r, q_r)\) exists within the Twin Prime set (\(TP\)), we apply arbitrary even shifts \(2u\) across the infinite domain of natural numbers \(\mathbb{N}_{+}\). By leveraging the infinitude of primes, we analyze the structural branching behavior of the shifted elements. The framework shows that for select intervals, the shifted coordinate paths split dynamically into guaranteed composite odd tracks and pure prime tracks. When a prime track occurs, it yields a valid pair \((p_r+2u, q_r+2u) \in TP\), directly violating the initial boundary constraint of a final terminal pair. This contradiction shows that the generation of twin prime coordinate sets mirrors the infinite nature of \(\mathbb{N}_{+}\), concluding that the Twin Prime set is strictly infinite.
Author: Taha Muhammad
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225632327
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