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ON A PROBLEM OF RICHARD GUY

Published online by Cambridge University Press:  13 September 2021

NGUYEN XUAN THO*
Affiliation:
School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, Hanoi, Vietnam

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 $2\nmid a+b$ .

Type
Research Article
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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