Algorithmic determination of a large integer in the two-term Machin-like formula for pi

Sanjar M. Abrarov; Rajinder K. Jagpal; Rehan Siddiqui; Brendan M. Quine

doi:10.3390/math9172162

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arXiv · 2107.01027

Algorithmic determination of a large integer in the two-term Machin-like formula for pi

Sanjar M. Abrarov·Rajinder K. Jagpal·Rehan Siddiqui·Brendan M. Quine

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Abstract

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.

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BibTeX RIS

Sanjar M. Abrarov, Rajinder K. Jagpal, Rehan Siddiqui, Brendan M. Quine. 2021-06-30. Algorithmic determination of a large integer in the two-term Machin-like formula for pi. https://doi.org/10.3390/math9172162

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