A symmetric Diophantine equation involving biquadrates

Ajai Choudhry
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 24, 2018, Number 2, Pages 140–144
DOI: 10.7546/nntdm.2018.24.2.140-144
Full paper (PDF, 121 Kb)

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Authors and affiliations

Ajai Choudhry
13/4 A Clay Square, Lucknow – 226001, India

Abstract

This paper is concerned with the Diophantine equation

∑_i=1ⁿ a_ix_i⁴ = ∑_i=1ⁿ a_iy_i⁴,where n ≥ 3 and a_i, i = 1, 2, …, n, are arbitrary nonzero integers. While a method of obtaining numerical solutions of such an equation has recently been given, it seems that an explicit parametric solution of this Diophantine equation has not yet been published. We obtain a multi-parameter solution of this equation for arbitrary values of a_i and for any positive integer n ≥ 3, and deduce specific solutions when n = 3 and n = 4. The numerical solutions thus obtained are much smaller than the integer solutions of such equations obtained earlier.

Keywords

• Biquadrates
• Fourth powers
• Quartic Diophantine equation

2010 Mathematics Subject Classification

• 11D25

References

1. Choudhry, A. (1991) Symmetric Diophantine systems, Acta Arithmetica, 59, 291–307.
2. Choudhry, A. (2013) Equal sums of like powers and equal products of integers, Rocky Mountain Journal of Mathematics, 43, 763–792
3. Dickson, L. E. (1992) History of theory of numbers, Vol. 2, Chelsea Publishing Company, New York.
4. Izadi, F., & Baghalaghdam, M. (2017) On the Diophantine equation ∑_i=1ⁿ a_ix_i⁴ = ∑_j=1ⁿ a_jy_j⁴, available at arXiv:1701.02605.