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

TL;DR

The paper develops a framework of higher-dimensional visual proofs for Faulhaber polynomials by dissecting pyramidal sums into lower-dimensional sections. It presents a 4D visual proof of Nicomachus' Theorem for sums of cubes, derived through 3D and 2D sections and a truncation lemma. It then extends the method to sums of fourth powers, revealing the factorization $\sum_{k=1}^n k^4 = \frac{n(n+1)(2n+1)(3n^2+3n-1)}{30}$ and explaining the appearance of the irreducible factor $3n^2+3n-1$ via a five-pyramid DIY construction across 5D, isolating subfactors $n(n+1)$ and $n+\tfrac{1}{2}$ and a root-based cut. The results provide geometric intuition for Faulhaber polynomials and their factor structures in higher dimensions.

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.

Figures (44)

• Figure 1: Sum of consecutive odd numbers
• Figure 2: Triangular numbers
• Figure 3: Visual proof for the formula for triangular numbers
• Figure 4: Visual proof of Archimedes’ identity
• Figure 5: Visual proof of Nicomachus’ Theorem

Theorems & Definitions (8)

• Remark 1.1: Previous Work on Visual Proofs in Higher Dimensions
• Lemma 2.1
• proof
• Remark 2.2
• Remark 2.3: Algebra vs. Visual Arguments I
• Remark 4.1: Algebra vs. Visual Arguments II
• Lemma 4.2
• proof