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

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 $$\frac{\pi }{4}=2^{k-1}\arctan \left(\frac{1}{u_{1,k}}\right)+\arctan \left(\frac{1}{u_{2,k}}\right),k\in \mathbb{Z},k\ge 1,$$ where $u_{1,k}$ can be chosen as an integer $u_{1,k}=\lfloor a_k/\sqrt{2-a_{k-1}}\rfloor$ 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.

arXiv comment30 pages

References in corpus (5)

• A formula for pi involving nested radicals
• An iteration procedure for a two-term Machin-like formula for pi with small Lehmer's measure
• Sinc integrals and tiny numbers
• Unconditional applicability of Lehmer's measure to the two-term Machin-like formula for $\pi$
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Cited by in corpus (2)

• An iterative method for computing $\pi$ by argument reduction of the tangent function
• A rational approximation of the two-term Machin-like formula for $\pi$