Algorithmic determination of a large integer in the two-term Machin-like formula for pi
| Autor: | Abrarov, Sanjar M., Jagpal, Rajinder K., Siddiqui, Rehan, Quine, Brendan M. |
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| Rok vydání: | 2021 |
| Předmět: | |
| Zdroj: | Mathematics 2021, 9(17), 2162 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.3390/math9172162 |
| Popis: | In our earlier publication we have shown how to compute by iteration a rational number ${u_{2,k}}$ in the two-term Machin-like formula for pi of kind $$\frac{\pi}{4}=2^{k-1}\arctan\left(\frac{1}{u_{1,k}}\right)+\arctan\left(\frac{1}{u_{2,k}}\right),\qquad k\in \mathbb{Z},\quad k\ge 1,$$ where ${u_{1,k}}$ can be chosen as an integer ${u_{1,k}} = \left\lfloor{{a_k}/\sqrt{2-a_{k-1}}}\right\rfloor$ with nested radicals defined as ${a_k}=\sqrt{2+a_{k-1}}$ and $a_0 = 0$. In this work we report an alternative method for determination of the integer $u_{1,k}$. This approach is based on a simple iteration and does not require any irrational (surd) numbers from the set $\left\{a_k\right\}$ in computation of the integer $u_{1,k}$. Mathematica programs validating these results are presented. Comment: 30 pages |
| Databáze: | arXiv |
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