A Balanced Three-term Generalization of Nicomachus' Identity
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866909913406504960 |
|---|---|
| author | Kim, Seon-Hong Stolarsky, Kenneth B. |
| author_facet | Kim, Seon-Hong Stolarsky, Kenneth B. |
| contents | We present a generalization of the classical Nicomachus' identity for the sum of the first $n$ cubes. Unlike previous generalizations, it has three rather than two terms, and involves not just one, but two distinct triangular numbers, and each term is of degree $4$ in $\lfloor n/2 \rfloor$. The asymptotic behavior for large $n$ leads to continued fractions with remarkable (but conjectural) properties. Moreover, we give a way of looking at squares of triangular numbers that involves the square root of $11$ and show it is a limiting case of a non-obvious identity involving truncations of the continued fraction expansion of that square root. The details involve a nonlinear recurrence that (with appropriate initial conditions) unexpectedly produces only integers, a ``Somos-type'' phenomenon. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_15133 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Balanced Three-term Generalization of Nicomachus' Identity Kim, Seon-Hong Stolarsky, Kenneth B. Number Theory Primary 11B83, Secondary 11D25 We present a generalization of the classical Nicomachus' identity for the sum of the first $n$ cubes. Unlike previous generalizations, it has three rather than two terms, and involves not just one, but two distinct triangular numbers, and each term is of degree $4$ in $\lfloor n/2 \rfloor$. The asymptotic behavior for large $n$ leads to continued fractions with remarkable (but conjectural) properties. Moreover, we give a way of looking at squares of triangular numbers that involves the square root of $11$ and show it is a limiting case of a non-obvious identity involving truncations of the continued fraction expansion of that square root. The details involve a nonlinear recurrence that (with appropriate initial conditions) unexpectedly produces only integers, a ``Somos-type'' phenomenon. |
| title | A Balanced Three-term Generalization of Nicomachus' Identity |
| topic | Number Theory Primary 11B83, Secondary 11D25 |
| url | https://arxiv.org/abs/2511.15133 |