Sum of Three Squares and Particle in a Cubic Box - Degeneracies
Abstract
We address the problem of determining the degeneracy of a particular energy eigenvalue corresponding to a quantum particle trapped in a cubic box. In other words, we aim to find the number of ways a specific positive integer can be written as the sum of three natural number squares. This is a classic problem in number theory. We provide a closed-form, exact solution to the problem. We do not claim to provide proofs, as we have none; instead, we build up the formulae starting from simple cases where the answer could be deduced by studying the list of degeneracies. We present ample examples that could be verified against the degeneracy calculated numerically via the brute-force method. We arrived at the solution by meticulously studying the list of degeneracies for the first five million numbers, generated by MathematicacircledR.
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BibTeX
@misc{khastgir2026three,
title = {{Sum of Three Squares and Particle in a Cubic Box - Degeneracies}},
author = {S. Pratik Khastgir},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23794},
eprint = {2609.23794},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23794},
}
RIS
TY - UNPB
TI - Sum of Three Squares and Particle in a Cubic Box - Degeneracies
AU - S. Pratik Khastgir
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23794
UR - https://arxiv.org/abs/2609.23794
AB - We address the problem of determining the degeneracy of a particular energy eigenvalue corresponding to a quantum particle trapped in a cubic box. In other words, we aim to find the number of ways a specific positive integer can be written as the sum of three natural number squares. This is a classic problem in number theory. We provide a closed-form, exact solution to the problem. We do not claim to provide proofs, as we have none; instead, we build up the formulae starting from simple cases where the answer could be deduced by studying the list of degeneracies. We present ample examples that could be verified against the degeneracy calculated numerically via the brute-force method. We arrived at the solution by meticulously studying the list of degeneracies for the first five million numbers, generated by Mathematica^circledR.
ER -