Nontrivial Solutions to a Cubic Identity and the Factorization of n²+n+1

We investigate a variation of Nicomachus's identity in which one term in the cubic sum is replaced by a different cube. Specifically, we study the Diophantine identity \[ ∑ⱼ₌₁ⁿ j^3 + x^3 - k^3 = ( ∑ⱼ₌₁ⁿ j + x - k )^2 \] and classify all integer solutions $(k,x,n)$. A full parametric family of nontrivial solutions was introduced in a 2005 paper, along with a conjectural condition for when such solutions exist. We provide a complete proof of this characterization and show it is equivalent to a structural condition on the prime factorization of n² + n + 1. Our argument connects this identity to classical results in the theory of binary quadratic forms. In particular, we analyze the equation a² + ab + b² = n² + n + 1, interpreting it as a norm in the ring of Eisenstein integers Z[ω], where ω= 1 + √-32. This yields a surprising connection between a modified combinatorial identity and the arithmetic of algebraic number fields.