A remark on the Diophantine equation (x^3-1)/(x-1)=(y^n-1)/(y-1)

He, Bing

doi:10.3336/gm.44.1.01

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2023

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Cited by 6 publications

(3 citation statements)

References 20 publications

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“…The above conjecture is a very difficult problem in Diophantine equations. It was solved for some special cases (see [3, 5, 6, 9, 10, 12, 14–18, 22, 23, 26, 29–37]). But, in general, the problem is far from solved.…”

Section: Introductionmentioning

confidence: 99%

A Note on the Goormaghtigh Equation Concerning Difference Sets

Fujita

Mao-hua²

2023

Bull. Aust. Math. Soc.

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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.

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Section: Introductionmentioning

confidence: 99%

A Note on the Goormaghtigh Equation Concerning Difference Sets

Fujita

Mao-hua²

2023

Bull. Aust. Math. Soc.

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“…A good survey of results up to 2001 or so can be found in the paper of Shorey [24]. For more recent work, the reader may consult [11], [12], [17], [18], [19] and [30]. The current state of the art for applying techniques from Diophantine Approximation to (1) can be found in [4], and in the papers of Nesterenko and Shorey [21], and of Bugeaud and Shorey [7].…”

Section: Introductionmentioning

confidence: 99%

Goormaghtigh's equation: small parameters

Bennett¹,

Garbuz²,

Marten³

2020

Publ. Math. Debrecen

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“…Equation (1) has been studied extensively under the condition that x > y > 1 (see [1], [3], [4], [5], and [6], to give just a few examples). The only known solutions (of the form) (x, y, m, n) under this condition are (5,2,3,5) and (90, 2, 3, 13). P. Yuan [8] considered the special case of equation 1given by…”

Section: Introductionmentioning

confidence: 99%