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On an elementary method for solving $Ax^4-By^2=1$

P. G. Walsh

Abstract

A new method for solving quartic equations due to Luo and Lin is investigated both computationally and theoretically. As a result, a completely straightforward elementary method is given for solving Bumby's equation $3X^4-2Y^2=1$, along with a conjecture, which if resolved, would enable a similar proof for a possibly infinite family of similar equations.

On an elementary method for solving $Ax^4-By^2=1$

Abstract

A new method for solving quartic equations due to Luo and Lin is investigated both computationally and theoretically. As a result, a completely straightforward elementary method is given for solving Bumby's equation , along with a conjecture, which if resolved, would enable a similar proof for a possibly infinite family of similar equations.
Paper Structure (7 sections, 1 theorem, 20 equations)

This paper contains 7 sections, 1 theorem, 20 equations.

Table of Contents

  1. Introduction
  2. Solving Bumby's equation $3X^4-2Y^2=1$
  3. The Factor Base
  4. The Main Case: $n\equiv 1 \; (\bmod \; 840)$
  5. Completion of the Proof
  6. The Extent of the Method
  7. Computational Results Pertaining to the Method

Key Result

Proposition 1

If $P_n=x^2$ for some integer $x$, then $n \equiv \pm 1, \pm 3 \; (\bmod \; 840)$.

Theorems & Definitions (1)

  • Proposition 1