Three essays on Machin's type formulas

TL;DR

This paper investigates Machin's type formulas, providing a complete classification of two-term formulas with specific properties, a method to generate infinitely many formulas, and corrections to previous results involving the golden section.

Contribution

It offers a comprehensive classification of two-term Machin formulas with 2-integer arctangents, introduces a method to find infinitely many formulas, and corrects earlier oversights involving the golden section.

Findings

All two-term Machin formulas with 2-integer arctangent functions are characterized.

A method to generate infinitely many Machin formulas with N terms is developed.

All two-term formulas involving arctangents of powers of the golden section are identified.

Abstract

We study three questions related to Machin's type formulas. The first one gives all two terms Machin formulas where both arctangent functions are evaluated $2$-integers, that is values of the form $b/2^a$ for some integers $a$ and~$b$. These formulas are computationally useful because multiplication or division by a power of two is a very fast operation for most computers. The second one presents a method for finding infinitely many formulas with $N$ terms. In the particular case $N=2$ the method is quite useful. It recovers most known formulas, gives some new ones, and allows to prove in an easy way that there are two terms Machin formulas with Lehmer measure as small as desired. Finally, we correct an oversight from previous result and give all Machin's type formulas with two terms involving arctangents of powers of the golden section.