Nontrivial Solutions to a Cubic Identity and the Factorization of $n^2+n+1$
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914339045244928 |
|---|---|
| author | Fejzić, Hajrudin |
| author_facet | Fejzić, Hajrudin |
| contents | 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 \[ \sum_{j=1}^{n} j^3 + x^3 - k^3 = \left( \sum_{j=1}^{n} j + x - k \right)^2 \] 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 $ n^2 + n + 1 $.
Our argument connects this identity to classical results in the theory of binary quadratic forms. In particular, we analyze the equation $a^2 + ab + b^2 = n^2 + n + 1$, interpreting it as a norm in the ring of Eisenstein integers $\mathbb{Z}[ω]$, where $ω= \frac{1 + \sqrt{-3}}{2}$. This yields a surprising connection between a modified combinatorial identity and the arithmetic of algebraic number fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_14937 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nontrivial Solutions to a Cubic Identity and the Factorization of $n^2+n+1$ Fejzić, Hajrudin General Mathematics Primary 11D09, Secondary 11R04, 11A41 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 \[ \sum_{j=1}^{n} j^3 + x^3 - k^3 = \left( \sum_{j=1}^{n} j + x - k \right)^2 \] 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 $ n^2 + n + 1 $. Our argument connects this identity to classical results in the theory of binary quadratic forms. In particular, we analyze the equation $a^2 + ab + b^2 = n^2 + n + 1$, interpreting it as a norm in the ring of Eisenstein integers $\mathbb{Z}[ω]$, where $ω= \frac{1 + \sqrt{-3}}{2}$. This yields a surprising connection between a modified combinatorial identity and the arithmetic of algebraic number fields. |
| title | Nontrivial Solutions to a Cubic Identity and the Factorization of $n^2+n+1$ |
| topic | General Mathematics Primary 11D09, Secondary 11R04, 11A41 |
| url | https://arxiv.org/abs/2508.14937 |