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ON A PROBLEM OF RICHARD GUY

Part of: Diophantine equations

Published online by Cambridge University Press:  13 September 2021

NGUYEN XUAN THO Open the ORCID record for NGUYEN XUAN THO [Opens in a new window]
NGUYEN XUAN THO*
Affiliation:
School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, Hanoi, Vietnam
*
e-mail: tho.nguyenxuan1@hust.edu.vn
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Abstract

In the 1993 Western Number Theory Conference, Richard Guy proposed Problem 93:31, which asks for integers n representable by ${(x+y+z)^3}/{xyz}$, where $x,\,y,\,z$ are integers, preferably with positive integer solutions. We show that the representation $n={(x+y+z)^3}/{xyz}$ is impossible in positive integers $x,\,y,\,z$ if $n=4^{k}(a^2+b^2)$, where $k,\,a,\,b\in \mathbb {Z}^{+}$ are such that $k\geq 3$ and $2\nmid a+b$.

Keywords

cubic equationsp-adic numbersHilbert symbols

MSC classification

Primary: 11D25: Cubic and quartic equations
Secondary: 11D88: $p$-adic and power series fields

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 105 , Issue 1 , February 2022 , pp. 12 - 18
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Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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Footnotes

The author is supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) (grant number 10.04-2019.314).

References

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