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FROBENIUS NUMBERS ASSOCIATED WITH DIOPHANTINE TRIPLES OF $x^2+y^2=z^3$

Part of: Diophantine equations Combinatorics Semigroups

Published online by Cambridge University Press:  14 November 2024

TAKAO KOMATSU Open the ORCID record for TAKAO KOMATSU [Opens in a new window] ,
NEHA GUPTA Open the ORCID record for NEHA GUPTA [Opens in a new window]  and
MANOJ UPRETI Open the ORCID record for MANOJ UPRETI [Opens in a new window]
TAKAO KOMATSU*
Affiliation:
Faculty of Education, Nagasaki University, Nagasaki 852-8521, Japan
NEHA GUPTA
Affiliation:
Department of Mathematics, School of Natural Sciences, Shiv Nadar Institute of Eminence, Gautam Buddha Nagar, Greater Noida 201314, India e-mail: neha.gupta@snu.edu.in
MANOJ UPRETI
Affiliation:
Department of Mathematics, School of Natural Sciences, Shiv Nadar Institute of Eminence, Gautam Buddha Nagar, Greater Noida 201314, India e-mail: mu506@snu.edu.in
*
e-mail: komatsu@nagasaki-u.ac.jp
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Abstract

We give an explicit formula for the Frobenius number of triples associated with the Diophantine equation $x^2+y^2=z^3$, that is, the largest positive integer that can only be represented in p ways by combining the three integers of the solutions of $x^2+y^2=z^3$. For the equation $x^2+y^2=z^2$, the Frobenius number has already been given. Our approach can be extended to the general equation $x^2+y^2=z^r$ for $r>3$.

Keywords

Frobenius problemDiophantine equationPythagorean tripleApéry set

MSC classification

Primary: 11D07: The Frobenius problem
Secondary: 05A15: Exact enumeration problems, generating functions 11D04: Linear equations 11D25: Cubic and quartic equations 20M14: Commutative semigroups

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 112 , Issue 1 , August 2025 , pp. 57 - 66
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The first author was supported by JSPS KAKENHI Grant Number 24K22835.

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