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The diophantine equation $x^4+y^4=z^4+w^4$

Ajai Choudhry, Arman Shamsi Zargar

Abstract

Since 1772, when Euler first described two methods of obtaining two pairs of biquadrates with equal sums, several methods of solving the diophantine equation $x^4+y^4=z^4+w^4$ have been published. All these methods yield parametric solutions in terms of homogeneous bivariate polynomials of odd degrees. In this paper we describe a method that yields three parametric solutions of the aforesaid diophantine equation in terms of homogeneous bivariate polynomials of even degrees, namely degrees~$74$, $88$ and $132$ respectively.

The diophantine equation $x^4+y^4=z^4+w^4$

Abstract

Since 1772, when Euler first described two methods of obtaining two pairs of biquadrates with equal sums, several methods of solving the diophantine equation have been published. All these methods yield parametric solutions in terms of homogeneous bivariate polynomials of odd degrees. In this paper we describe a method that yields three parametric solutions of the aforesaid diophantine equation in terms of homogeneous bivariate polynomials of even degrees, namely degrees~, and respectively.
Paper Structure (4 sections, 20 equations)

This paper contains 4 sections, 20 equations.

Table of Contents

  1. Introduction
  2. Generating new parametric solutions from old
  3. Even degree parametric solutions of Eq. \ref{['quartic']}
  4. Some open problems