On a BBP-type formula for $π^2$ in the golden ratio base
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866913976912183296 |
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| author | Cloitre, Benoit |
| author_facet | Cloitre, Benoit |
| contents | This paper presents a detailed, self-contained proof of a BBP-type formula for $π^2$ expressed in the golden ratio base, $ϕ$. The formula was discovered empirically by the author in 2004. The proof presented herein is built upon a fundamental geometric identity connecting $ϕ$ to the fifth roots of unity, offering an intuitive and direct path to the result. The power of the underlying methodology is then demonstrated by extending it to establish a new, computationally efficient Machin-like formula for $ζ(3)$, expressed through rapidly converging, hierarchical series involving the golden ratio. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_03743 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a BBP-type formula for $π^2$ in the golden ratio base Cloitre, Benoit General Mathematics 11Y60 (Primary), 33B30, 11M06, 40A25 (Secondary) This paper presents a detailed, self-contained proof of a BBP-type formula for $π^2$ expressed in the golden ratio base, $ϕ$. The formula was discovered empirically by the author in 2004. The proof presented herein is built upon a fundamental geometric identity connecting $ϕ$ to the fifth roots of unity, offering an intuitive and direct path to the result. The power of the underlying methodology is then demonstrated by extending it to establish a new, computationally efficient Machin-like formula for $ζ(3)$, expressed through rapidly converging, hierarchical series involving the golden ratio. |
| title | On a BBP-type formula for $π^2$ in the golden ratio base |
| topic | General Mathematics 11Y60 (Primary), 33B30, 11M06, 40A25 (Secondary) |
| url | https://arxiv.org/abs/2508.03743 |