Geometric Insights into the Goldbach Conjecture

The binary Goldbach conjecture states that every even integer greater than 2 is the sum of two primes. We analyze a variant of this conjecture, positing that every even integer 2N ≥ 8 is the sum of two distinct primes P and Q. We establish a novel equivalence between this statement and a geometric construction: the conjecture holds if and only if for every N ≥ 4, there exists an integer M ∈ [1, N − 3] such that the L-shaped region N2 − M2 (between nested squares) has a semiprime area P · Q, where P = N − M and Q = N + M. We define the set DN = n Q−P 2 2 < P < N < Q < 2N, P, Q prime o of half-differences arising from prime pairs straddling N with Q < 2N. The conjecture is equivalent to the non-emptiness of DN ∩ {N − p | 3 ≤ p < N, p prime}. We conduct a computational analysis for N ≤ 214 and define a gap function G(N) = log2(2N) − ((N − 3) − |DN|). Our experimental results show that the minimum of G(N) is positive and increasing across intervals [2m, 2m+1]. This result, G(N) > 0, establishes that |DN| > (N − 3) − log2(2N). Under this bound, the pigeonhole principle applied to the cardinality of the candidate set {N − p | 3 ≤ p < N, p prime} (of size π(N − 1) − 1) and the bad positions (of size (N − 3) − |DN| < log2(2N)) implies a non-empty intersection for all N ≥ 4, yielding a proof of the conjecture. Our work establishes a novel geometric framework and demonstrates its viability through extensive computation.