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

TL;DR

This paper introduces a new iterative method to determine an integer in a Machin-like formula for pi, simplifying calculations by avoiding irrational numbers used in previous approaches.

Contribution

The authors propose an alternative iterative approach to compute the integer in the two-term Machin-like formula for pi, eliminating the need for nested radicals.

Findings

The new method accurately computes the integer without irrational numbers.

Mathematica programs validate the iterative approach.

The approach simplifies the computation process for the Machin-like formula.

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.