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A NOTE ON THE GOORMAGHTIGH EQUATION CONCERNING DIFFERENCE SETS
Published online by Cambridge University Press: 23 June 2023
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
MSC classification
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- Research Article
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- Copyright
- © The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
References
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