A new form of the Machin-like formula for pi by iteration with increasing integers

TL;DR

This paper introduces a novel iterative Machin-like formula for pi that employs increasing integers, potentially enhancing computational efficiency, with tests showing stable Lehmer measures after multiple iterations.

Contribution

It proposes a new iterative Machin-like formula for pi that uses increasing integers, demonstrating stable Lehmer measures over multiple iterations.

Findings

Lehmer measure remains small for k ≥ 17

Lehmer measure does not significantly increase after 18 iterations

Potential for efficient pi computation using the new formula

Abstract

We present a new form of the Machin-like formula for $\pi$ that can be generated by using iteration. This form of the Machin-like formula may be promising for computation of the constant $\pi$ due to rapidly increasing integers at each step of the iteration. The computational test we performed shows that, with an integer $k \ge 17$, the Lehmer measure remains small and practically does not increase after $18$ steps of iteration.