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Three-term Machin-type formulae

Tomohiro Yamada

Abstract

We shall show that there exist only finitely many nondegenerate three-term Machin-type formulae and give explicit upper bounds for the sizes of variables.

Three-term Machin-type formulae

Abstract

We shall show that there exist only finitely many nondegenerate three-term Machin-type formulae and give explicit upper bounds for the sizes of variables.

Paper Structure

This paper contains 5 sections, 8 theorems, 156 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A lower bound for linear forms of three logarithms
  4. Upper bounds for exponents
  5. Proof of the Theorem

Key Result

Theorem 1.1

Assume that $x_1, x_2, x_3, y_1, y_2, y_3$ and $r$ are nonzero integers with $x_1, x_2, x_3>1$, $\{x_1, x_2, x_3\}\neq \{2, 3, 7\}$ satisfying eq11, and $\gcd(y_1, y_2, y_3)=1$ and $m_1, m_2, s_i, k_i, \ell_i (i=1, 2, 3)$ are corresponding integers with $m_2>m_1>0$ satisfying eq13. More detailed estimates are given in Tables tbl5 and tbl6.

Theorems & Definitions (15)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • ...and 5 more