Structural Number Theory: Research on Prime Number Rules and Unified Proofs of Several Number‑Theoretic Conjectures – An Original Theory and Complete Demonstration Based on Structural Covering (Series 3: Proofs of the Mersenne‑Prime Conjecture and Legendre's Conjecture)

Based on the Structural Covering theory, this paper constructs two types of number‑set correspondences. Starting from the maximum set, it computes either the target set or the minimal set for different propositions. By establishing the reductio‑ad‑absurdum hypothesis, screening redundant elements and computing the union of sets, a logical contradiction is further derived. This paper simultaneously proves two number‑theoretic conjectures: the Mersenne‑prime conjecture, which states that there exist infinitely many primes p such that 2^p‑1 is prime; and Legendre’s conjecture, which states that there always exists a prime between any two consecutive perfect squares.