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A NOTE ON THE GOORMAGHTIGH EQUATION CONCERNING DIFFERENCE SETS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 109 / Issue 3 / June 2024
- Published online by Cambridge University Press:
- 23 June 2023, pp. 443-452
- Print publication:
- June 2024
-
- Article
- Export citation
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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.
