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Three essays on Machin's type formulas

Armengol Gasull, Florian Luca, Juan L. Varona

TL;DR

This work advances the theory of Machin-type pi representations by delivering an exhaustive classification for two-term formulas with arctan arguments constrained to powers of two, introducing a constructive framework (the Machin's formulas machine) for generating Machin-like identities with arbitrary term counts, and clarifying identities involving powers of the golden section. It combines diophantine, Gaussian-integer, and algebraic-number techniques to enumerate sporadic and parametric families and to prove the existence of arbitrarily small Lehmer-measure formulas. The approach unifies and extends prior results (including ABCM and LuSt corrections), while also providing practical methods for computing pi via rapidly convergent arctan series anchored by carefully chosen rational arguments. The golden-section analysis yields a complete, corrected set of sixteen fundamental identities, refining the landscape of arctan-based pi formulas with algebraic-number structure. Overall, the paper both sharpens theoretical understanding and offers explicit, efficiently computable identities with modern computational relevance.

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.

Three essays on Machin's type formulas

TL;DR

This work advances the theory of Machin-type pi representations by delivering an exhaustive classification for two-term formulas with arctan arguments constrained to powers of two, introducing a constructive framework (the Machin's formulas machine) for generating Machin-like identities with arbitrary term counts, and clarifying identities involving powers of the golden section. It combines diophantine, Gaussian-integer, and algebraic-number techniques to enumerate sporadic and parametric families and to prove the existence of arbitrarily small Lehmer-measure formulas. The approach unifies and extends prior results (including ABCM and LuSt corrections), while also providing practical methods for computing pi via rapidly convergent arctan series anchored by carefully chosen rational arguments. The golden-section analysis yields a complete, corrected set of sixteen fundamental identities, refining the landscape of arctan-based pi formulas with algebraic-number structure. Overall, the paper both sharpens theoretical understanding and offers explicit, efficiently computable identities with modern computational relevance.

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 -integers, that is values of the form for some integers and~. 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 terms. In the particular case 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.
Paper Structure (13 sections, 4 theorems, 125 equations, 3 figures, 2 tables)

This paper contains 13 sections, 4 theorems, 125 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Machin's formulas with powers of two
  3. A reformulation
  4. Proof of Theorem \ref{['theo:power2']}
  5. The case $a_1\ge 1$
  6. The case $a_1=0$
  7. The Machin's formulas machine
  8. The functions $R_j(n,x)$
  9. Some properties of the functions $R_j(n,x)$
  10. Machin-like formulas associated to $R_j(n,x)$
  11. Some examples
  12. Machin-like identities with small Lehmer measure
  13. Machin's formulas with powers of the golden section

Key Result

Theorem 1

All solutions $(x_1,z_1,x_2,z_2)$ of equation eq:0 in non-zero rational numbers $x_1,x_2$ and rational numbers $z_1<z_2$ in $(0,1)$ of the form $2^{a_k}/b_k$ or $b_k/2^{a_k}$ for $k=1,2$ are the following ten sporadic ones together with the two parametric families

Figures (3)

  • Figure 1: The function $\frac{4}{\pi} (4\arctan(R_3(3,x)) - 3\arctan(R_0(4,x)))$, for $x \in (-5,5)$.
  • Figure 2: The function $\frac{4}{\pi} (7\arctan(R_3(13,x)) - 13\arctan(R_0(7,x)))$, for $x \in (-5,5)$.
  • Figure 3: The function $\frac{4}{\pi} (\frac{4}{13}\arctan(R_3(13,x)) - \frac{9}{7}\arctan(R_0(7,x)) + \frac{5}{8}\arctan(R_1(8,x)))$, for $x \in (-5,5)$.

Theorems & Definitions (7)

  • Theorem 1
  • Remark
  • Theorem 2
  • Theorem 3
  • proof
  • Corollary 4
  • proof