An Elementary Symmetric Polynomial Approach to the Proof of Goldbach's Conjecture and Legendre's Conjecture

We present a novel sieve framework based on elementary symmetric polynomials to study the existence and distribution of prime pairs. By introducing a “resonance elimination” mechanism (Proposition 5.2) into the error term of the inclusion-exclusion expansion, we successfully control the combinatorial explosion that plagues traditional sieve methods, obtaining optimal error estimates. Building upon this framework, we rigorously prove the following results: (1) The Symmetric Theorem: For every sufficiently large integer x, there exists an integer y such that x − y and x + y are both prime. This directly implies the even Goldbach conjecture. (2) Legendre’s Conjecture: For every sufficiently large integer n, there exists at least one prime between n 2 and (n + 1)2 . Our methods are entirely based on elementary number theory and classical Fourier analysis, requiring no unproven hypotheses about the zeros of the ζ-function or the theory of automorphic forms. The framework provides a self-contained resolution of these classical problems in additive number theory.