On refinements of two-term Machin-like formulas
Bakir Farhi
TL;DR
The paper addresses refining two-term Machin-like formulas by exploiting the continued fraction of the arctan-argument ratio $\alpha = \dfrac{\arctan{u_0}}{\arctan{u_1}}$. A CF-based framework yields a decreasing sequence of rational arguments $u_n$ and integers $a_{-n}$ with $a_{-n} \arctan{u_n} + a_{-n+1} \arctan{u_{n+1}} = \dfrac{\pi}{4}$, and closed forms for $a_{-n-1}$ and $\arctan{u_n}$ in terms of convergents $N_n/D_n$. It is proved that $\alpha$ is irrational, $\arctan{u_n} \to 0$ with $a_{-n} u_n + a_{-n+1} u_{n+1} \to \dfrac{\pi}{4}$ at a geometric rate $O(1/D_{n-1}^2)$, and the Euler two-term example $\arctan(1/2) + \arctan(1/3) = \pi/4$ yields concrete refined identities and rapidly converging rational approximations to $\pi$. The results provide a systematic method for generating refined small-argument Machin-like formulas with potential for high-precision $\pi$ computations.
Abstract
We develop a refinement process for two-term Machin-like formulas: $a_0 \arctan{u_0} + a_1 \arctan{u_1} = \fracπ{4}$ (where $a_0 , a_1 \in \mathbb{Z}$, $u_0 , u_1 \in \mathbb{Q}_+^*$, $u_0 > u_1$) by exploiting the continued fraction expansion of the ratio $α:= \frac{\arctan{u_0}}{\arctan{u_1}}$. This construction yields a sequence of derived two-term Machin-like formulas: $a_{- n} \arctan{u_n} + a_{- n + 1} \arctan{u_{n + 1}} = \fracπ{4}$ ($n \in \mathbb{N}$) with positive rational arguments $u_n$ decreasing to zero and corresponding integer coefficients $a_{- n}$. We derive closed forms and estimates for $a_{-n}$ and $u_n$ in terms of the convergents of $α$ and prove that the associated rational sequence $(a_{- n} u_n + a_{- n + 1} u_{n + 1})_n$ converges to $π/4$ with geometric decay. The method is illustrated using Euler's two-term Machin-like formula : $\arctan(1/2) + \arctan(1/3) = π/4$.