On Representations of Even Integers as a Sum of Two Semiprimes

Representations of a large even integer N as a sum of two semiprimes (products of two primes, squares allowed) are studied. Using a smooth bilinear weight W localized on uv ≍ N and the Hardy–Littlewood circle method, an asymptotic formula of order N with a power saving in log N is obtained: \( R(N) \;=\; \sum_{\begin{array}{c}s_1+s_2=N \\ s_1,s_2\in S_2 \end{array}} W(s_1)W(s_2) \;=\; \mathfrak{S}(N)\,N \;+\; O\!\Big(\frac{N}{(\log N)^{1+\delta}}\Big) \) for some absolute δ > 0, where the multiplicative singular series \( \mathfrak{S}(N) \) is bounded and bounded away from zero along even N. The argument uses a bilinear Bombieri–Vinogradov type input (via the dispersion method and the multiplicative large sieve) together with minor-arc bounds based on Gallagher’s lemma. As a direct consequence, every sufficiently large even integer is the sum of two semiprimes (unweighted existence). A computational check (unweighted) finds no counterexamples up to N = 106 except 2, 4, 6, 22, in agreement with the asymptotic formula.