Some Diophantine equations concerning biquadrates

Ajai Choudhry
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 21, 2015, Number 4, Pages 1–5
Full paper (PDF, 130 Kb)

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Authors and affiliations

Ajai Choudhry
13/4 A Clay Square,
Lucknow, India

Abstract

This paper is concerned with integer solutions of the diophantine equation x₁⁴ + x₂⁴ + x₃⁴ = k x₄² where k is a given positive integer. Till now, integer and parametric solutions of this diophantine equation have been published only when k = 1 or 2 or 3. In this paper we obtain parametric solutions of this equation for 43 values of k ≤ 100. We also show that the equation cannot have any solution in integers for 54 values of k ≤ 100. The solvability of the equation x₁⁴ + x₂⁴ + x₃⁴ = k x₄² where k could not be determined for three values of k ≤ 100, namely 34, 35 and 65.

Keywords

• Biquadrates
• Sums of biquadrates

AMS Classification

• 11D25

References

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