A Computational and Analytic Investigation of Sun's (2-4-6-8) Binomial Representation Conjecture: Asymptotic Evidence, Local Theory, and Limitations

Abstract We investigate the conjecture of Sun Zhiwei that every non-negative integer n admits a representation n = C(x,2) + C(y,4) + C(z,6) + C(w,8), x, y, z, w ∈ ℤ≥0 where C(t,k) = t(t−1)···(t−k+1)/k! denotes the binomial coefficient. We establish three main results: Theorem A (Computational Verification). The conjecture holds for all 1 ≤ n ≤ 10⁶, with every such n admitting at least 2 representations. The minimum is exactly 2, attained at n ∈ {9590, 24935, 33845}, and the maximum is 568. Theorem B (Local Solubility). The representation set A satisfies A ≡ ℤ/mℤ for all moduli m ∈ {2,3,4,5,6,7,8,9,10,11,12,16,24,32,48,64}. There are no local obstructions to the conjecture. Theorem C (Circle Method Framework). We establish a rigorous Hardy–Littlewood circle method framework. The singular integral satisfies J(n) ≍ n^(1/24). The singular series 𝔖(n) > 0 for all n (conditional on local solubility). However, a complete proof of R(n) > 0 for all sufficiently large n faces a fundamental obstacle: the minor arc contribution cannot be controlled by current technology due to the high-degree polynomial C(w,8) having too few variables relative to its degree. Mathematical Status: The conjecture is strongly supported by computational and local evidence but remains open as a theorem. The primary obstruction is the minor arc problem for the degree-8 term with a single variable. 1. Introduction and Problem Classification 1.1 The Conjecture Conjecture (Sun Zhiwei). Every non-negative integer n can be written as n = C(x,2) + C(y,4) + C(z,6) + C(w,8) for some x, y, z, w ∈ ℤ≥0. The four polynomial sequences are: ● C(x,2) = x(x−1)/2: triangular numbers {0, 1, 3, 6, 10, 15, 21, …} ● C(y,4) = y(y−1)(y−2)(y−3)/24: values {0, 1, 5, 15, 35, 70, …} ● C(z,6) = z(z−1)···(z−5)/720: values {0, 1, 7, 28, 84, 210, …} ● C(w,8) = w(w−1)···(w−7)/40320: values {0, 1, 9, 45, 165, 495, …} 1.2 Problem Classification This problem belongs to Category D: mixed-degree polynomial representation problem, with elements of additive basis theory. It is not a classical Waring problem (which concerns sums of k-th powers xᵏ). The polynomials C(t,k) are of degree k but are not monomials; they are integer-valued polynomials with leading coefficient 1/k!. The most precise classification is a representation problem for a sum of four integer-valued polynomials of distinct even degrees 2, 4, 6, 8. This places it in the tradition of Hilbert–Kamke problems and Birch's theorem on forms, but with the crucial distinction that the polynomials are not homogeneous.