A Complete Proof of the Even Goldbach Conjecture, Legendre's Conjecture, and the Composite Factor Prime Representation Theorem via a Symmetric Sieve Method

We present a rigorous proof of the even Goldbach conjecture, Legendre’s conjecture, and the Composite Factor Prime Representation Theorem using a novel symmetric sieve framework. The method is based on constructing a “small prime sieve” via the Chinese Remainder Theorem, followed by a double averaging procedure and an inclusion-exclusion expansion over the large primes. The core technical innovation is a resonance elimination mechanism (Proposition 7) that decomposes character sums into sums of geometric series, effectively breaking potential resonances that would otherwise amplify the error terms. For small moduli (d ≤ p 1/3 ), we apply a double denominator estimate combined with Cauchy-Schwarz to bound the error by O(p 2/3+o(1)). For large moduli (d > p1/3 ), we employ Fourier expansion on the group Z/M0Z to obtain a uniform bound on the character sums, and then use the elementary symmetric polynomial method—connecting the inclusion-exclusion tail to a Poisson distribution via Maclaurin’s inequality—to bound the tail by O(p −1/3+o(1)). The total error is bounded by O(p 5/3+o(1)), which is negligible compared to the main term ∼ p 2/(log p) 2 . From this framework, we prove a Symmetric Theorem: for every sufficiently large integer x, there exists an integer y such that x − y and x + y are both prime. The even Goldbach conjecture follows immediately. Legendre’s conjecture is proved via a hierarchical covering argument. Finally, as a corollary of Goldbach, we prove the Composite Factor Prime Representation Theorem: for any composite N = m·n, N can be expressed as the sum of exactly m primes (and also n primes). All methods are elementary, relying only on classical Fourier analysis, the Chinese Remainder Theorem, Mertens’ theorems, and basic combinatorial inequalities. Comments and feedback are welcome. Please feel free to contact me via [email protected]