As one of the core unsolved problems in number theory, the Goldbach Conjecture proposes that every even integer greater than 2 can be expressed as the sum of two prime numbers. The Twin Prime Conjecture, another classic number theory problem, holds that there are infinitely many pairs of twin primes (prime pairs with a difference of 2). Based on the innovative theoretical system of "Shui's high-dimensional prime numbers + N-axis high-dimensional integration + self-similarity coefficient + Shui's constant" constructed in the previous study, this paper takes Shui's high-dimensional prime numbers and the 64-digit progressive cycle of π as the core anchors, first completes the rigorous proof of the Goldbach Conjecture, and then extends the theoretical system to realize the proof of the Twin Prime Conjecture. For the Goldbach Conjecture, we convert it into an equivalent proposition related to high-dimensional symmetry, derive the coupling formula of even number decomposition and π's 64-digit progressive cycle, and verify the conclusion through existence proof, validity verification and global traversal. For the Twin Prime Conjecture, we supplement the core symbols and theoretical extensions of twin primes, derive the high-dimensional interval constraint formula of twin prime pairs, and prove the infinity of twin prime pairs based on the progressive cycle consistency principle and the completeness of N-axis. This study extends the application of the theoretical system of Shui's high-dimensional prime numbers and the 64-digit progressive cycle of π, realizes the interdisciplinary integration of number theory, high-dimensional geometry and combinatorics, and provides a new rigorous proof path for two core problems of number theory.
Xiaogang shui (Mon,) studied this question.