2025/08/20 by Fejzić, Hajrudin
#11A41 #FOS: Mathematics #General Mathematics (math.GM) #Primary 11D09 #Secondary 11R04
paper · doi:10.48550/arxiv.2508.14937
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 j3 + x3 - k3 = ( ∑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 + √(-3))/(2). This yields a surprising connection between a modified combinatorial identity and the arithmetic of algebraic number fields.