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A NOTE ON THE GOORMAGHTIGH EQUATION CONCERNING DIFFERENCE SETS

Part of: Diophantine approximation, transcendental number theory Designs and configurations Diophantine equations

Published online by Cambridge University Press:  23 June 2023

YASUTSUGU FUJITA Open the ORCID record for YASUTSUGU FUJITA [Opens in a new window]  and
MAOHUA LE Open the ORCID record for MAOHUA LE [Opens in a new window]
YASUTSUGU FUJITA*
Affiliation:
Department of Mathematics, College of Industrial Technology, Nihon University, 2-11-1 Shin-ei, Narashino, Chiba, Japan
MAOHUA LE
Affiliation:
Institute of Mathematics, Lingnan Normal College, Zhanjiang, Guangdong 524048 China e-mail: [email protected]
*
e-mail: [email protected]
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Abstract

Let p be a prime and let r, s be positive integers. In this paper, we prove that the Goormaghtigh equation $(x^m-1)/(x-1)=(y^n-1)/(y-1)$, $x,y,m,n \in {\mathbb {N}}$, $\min \{x,y\}>1$, $\min \{m,n\}>2$ with $(x,y)=(p^r,p^s+1)$ has only one solution $(x,y,m,n)=(2,5,5,3)$. This result is related to the existence of some partial difference sets in combinatorics.

Keywords

exponential Diophantine equationGoormaghtigh equationgeneralised Ramanujan–Nagell equationBaker’s methodpartial difference set

MSC classification

Primary: 11D61: Exponential equations
Secondary: 05B10: Difference sets (number-theoretic, group-theoretic, etc.) 11J86: Linear forms in logarithms; Baker's method
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 109 , Issue 3 , June 2024 , pp. 443 - 452
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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