Geometric Proofs for Quadratic and Cubic Power Sums and a Reduction Algorithm for Higher Power Sums with Structural Links to Algebraic Integer Rings

Abstract

Power sum problem is a classical topic across combinatorial algebra and number theory, where traditional derivations mainly rely on Bernoulli numbers and generating function techniques, which lack intuitive geometric interpretations and straightforward decomposition method for higher-order sums. In this paper, we first propose novel geometric constructive proofs for quadratic and cubic power sums of the first n natural numbers, realizing the deduction of loworder power sum formulas via geometric partitioning rather than algebraic expansion. On this basis, we construct a pair of symmetric and antisymmetric auxiliary operators M(k) and K(k), and establish a new simple recursive decomposition algorithm that converts higher-order power sum expressions into combinations of lower-order power sums and polynomials containing n, achieving effective order reduction for odd powers without introducing Bernoulli numbers; for even powers the algorithm falls back to the standard binomial recursion. Furthermore, we explore the intrinsic structural correspondence between the proposed decomposition framework and algebraic integer rings: the symmetric-antisymmetric splitting of power sums is structurally parallel to the conjugate decomposition of Gaussian integers; the triple symmetry feature of cubic power sums parallels the structural properties of Eisenstein integers; meanwhile, the non-symmetric characteristic of higher-order decomposition can be extended to the non-commutative structure of Hurwitz quaternion integers. We conduct numerical verification for low-degree cases and compare our method with the classical Bernoulli number approach and explore class decomposition adopted in SPN architectures, illustrating the advantages of our algorithm in geometric intuition and iterative computation. We also research the calculation efficiency of the new algorithm. This work provides a new elementary decomposition paradigm for power sum research and simple intuitive proof for the sum of quadratic and cubic power, and establishes a feasible connection between power sum recursive splitting and algebraic integer ring theory