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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2509.19114 |
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| _version_ | 1866914060665094144 |
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| author | Alfano, Joseph Armstrong, Emily DeNolf, Vincent Sagarin, Jonah |
| author_facet | Alfano, Joseph Armstrong, Emily DeNolf, Vincent Sagarin, Jonah |
| contents | We present new combinatorial proofs of Nicomachus's Theorem for the sum of the third powers of the first n natural numbers. The key step is that we define a 4-dimensional block which comprises unit hyper-cubes. In our first proof we assemble 4 copies of this block to construct a rectangular solid. In our second proof we partition the block into two parts, then map and reassemble them into a solid whose shape is a step triangle along two coordinate axes, and is likewise on the other two. For corollaries we present a q-analogue of this identity using taxicab distance, and we interpret q-analogues from 4 different (sets of) authors, using taxicab distances from different starting points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_19114 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Interpreting a Sum of Third Powers by using a Geometric Assembly in Four Dimensions Alfano, Joseph Armstrong, Emily DeNolf, Vincent Sagarin, Jonah Combinatorics 05A19 We present new combinatorial proofs of Nicomachus's Theorem for the sum of the third powers of the first n natural numbers. The key step is that we define a 4-dimensional block which comprises unit hyper-cubes. In our first proof we assemble 4 copies of this block to construct a rectangular solid. In our second proof we partition the block into two parts, then map and reassemble them into a solid whose shape is a step triangle along two coordinate axes, and is likewise on the other two. For corollaries we present a q-analogue of this identity using taxicab distance, and we interpret q-analogues from 4 different (sets of) authors, using taxicab distances from different starting points. |
| title | Interpreting a Sum of Third Powers by using a Geometric Assembly in Four Dimensions |
| topic | Combinatorics 05A19 |
| url | https://arxiv.org/abs/2509.19114 |