On refinements of two-term Machin-like formulas
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914257099030528 |
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| author | Farhi, Bakir |
| author_facet | Farhi, Bakir |
| contents | We develop a refinement process for two-term Machin-like formulas: $a_0 \arctan{u_0} + a_1 \arctan{u_1} = \fracπ{4}$ (where $a_0 , a_1 \in \mathbb{Z}$, $u_0 , u_1 \in \mathbb{Q}_+^*$, $u_0 > u_1$) by exploiting the continued fraction expansion of the ratio $α:= \frac{\arctan{u_0}}{\arctan{u_1}}$. This construction yields a sequence of derived two-term Machin-like formulas: $a_{- n} \arctan{u_n} + a_{- n + 1} \arctan{u_{n + 1}} = \fracπ{4}$ ($n \in \mathbb{N}$) with positive rational arguments $u_n$ decreasing to zero and corresponding integer coefficients $a_{- n}$. We derive closed forms and estimates for $a_{-n}$ and $u_n$ in terms of the convergents of $α$ and prove that the associated rational sequence $(a_{- n} u_n + a_{- n + 1} u_{n + 1})_n$ converges to $π/4$ with geometric decay. The method is illustrated using Euler's two-term Machin-like formula : $\arctan(1/2) + \arctan(1/3) = π/4$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_10300 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On refinements of two-term Machin-like formulas Farhi, Bakir Number Theory Primary 11Y60, Secondary 11J70, 33B10, 11B37 We develop a refinement process for two-term Machin-like formulas: $a_0 \arctan{u_0} + a_1 \arctan{u_1} = \fracπ{4}$ (where $a_0 , a_1 \in \mathbb{Z}$, $u_0 , u_1 \in \mathbb{Q}_+^*$, $u_0 > u_1$) by exploiting the continued fraction expansion of the ratio $α:= \frac{\arctan{u_0}}{\arctan{u_1}}$. This construction yields a sequence of derived two-term Machin-like formulas: $a_{- n} \arctan{u_n} + a_{- n + 1} \arctan{u_{n + 1}} = \fracπ{4}$ ($n \in \mathbb{N}$) with positive rational arguments $u_n$ decreasing to zero and corresponding integer coefficients $a_{- n}$. We derive closed forms and estimates for $a_{-n}$ and $u_n$ in terms of the convergents of $α$ and prove that the associated rational sequence $(a_{- n} u_n + a_{- n + 1} u_{n + 1})_n$ converges to $π/4$ with geometric decay. The method is illustrated using Euler's two-term Machin-like formula : $\arctan(1/2) + \arctan(1/3) = π/4$. |
| title | On refinements of two-term Machin-like formulas |
| topic | Number Theory Primary 11Y60, Secondary 11J70, 33B10, 11B37 |
| url | https://arxiv.org/abs/2601.10300 |