Higher dimensional visual proofs, Nicomachus' 4D Theorem and the mysterious irreducible factor $(3n^2+3n-1)$ in the sum of fourth powers

TL;DR

This paper introduces a geometric approach to visual proofs of sum of powers formulas, extending Nicomachus's Theorem to four dimensions and explaining the emergence of a specific irreducible factor in higher dimensions.

Contribution

It presents a novel higher-dimensional visual proof technique for Faulhaber's formulas, including a four-dimensional proof of Nicomachus's Theorem and an explanation of a key irreducible factor in five dimensions.

Findings

Proved Nicomachus's Theorem in four dimensions.

Provided geometric interpretation of roots in sum of powers formulas.

Explained the appearance of the irreducible factor (3n^2+3n-1) in dimension five.

Abstract

Sums of powers $S_p(n)=\sum_{k=1}^n k^p$ can be described by Faulhaber's formula in terms of the Bernoulli numbers. The first cases of this formula admit visual proofs of various kinds, which lead to factorized Faulhaber polynomials. In this article we present a technique that yields higher-dimensional visual proofs for these factorized formulas, providing a geometric interpretation of the roots that appear. In particular, we prove Nicomachus's Theorem in four dimensions, and we visually explain the appearance, in dimension five, of the irreducible factor $(3n^2 +3n-1)$ in the polynomial ring over the rational numbers.