Geometric Insights into the Goldbach Conjecture

We prove that every even integer 2N ≥ 8 is the sum of two distinct primes. This variant of the classical Goldbach conjecture is established through three components: (1) a novel geometric equivalence reformulating the problem in terms of nested squares with semiprime areas, (2) a theoretical proof for all N ≥ 3275 using Dusart’s refinement on prime distribution, and (3) direct computational verification for 4 ≤ N ≤ 3274. The geometric framework reveals that the conjecture is equivalent to finding, for each N ≥ 4, an integer M ∈ [1, N − 3] such that the L-shaped region N2 − M2 between nested squares has area P · Q where P = N − M and Q = N + M are both prime. We define DN = {(Q − P)/2 | 2 < P < N < Q < 2N, both prime} to be the set of achievable half-differences from straddling prime pairs. The conjecture becomes equivalent to proving that DN ∩ {N − p | 3 ≤ p < N, p prime} ̸= ∅ for all N ≥ 4. Our gap function G(N) = log²(2N) − ((N − 3) − |DN|) measures the margin by which this condition holds. Computational analysis for N ∈ [4, 2¹⁴] reveals that G(N) > 0 universally, with minima strictly increasing across dyadic intervals. For N ≥ 3275, we prove theoretically that G(N) > 0 by showing that Dusart’s prime distribution theorem guarantees |DN| > (N − 3) − log²(2N). The pigeonhole principle then ensures existence of valid Goldbach partitions: since there are π(N −1)−1 > log²(2N) candidate primes P < N, and fewer than log²(2N) “bad” M-values, at least one candidate yields both P and Q = 2N − P prime. This completes the proof of the distinct-prime Goldbach variant and demonstrates the power of geometric reformulation combined with modern analytic number theory.