Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-05-20T09:05:24.890Z Has data issue: false hasContentIssue false
  • English
  • Français

A NOTE ON THE GOORMAGHTIGH EQUATION CONCERNING DIFFERENCE SETS

Part of: Designs and configurations Diophantine approximation, transcendental number theory 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: lemaohua2008@163.com
*
e-mail: fujita.yasutsugu@nihon-u.ac.jp
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abdollahi, A. and Mohammadi Hassanabadi, A., ‘Noncyclic graph of a group’, Comm. Algebra 35 (2007), 2057–2081.CrossRefGoogle Scholar
Baker, A. and Davenport, H., ‘The equations $3{x}^2-2={y}^2$ and $8{x}^2-7={z}^2$ ’, Q. J. Math. Oxford Ser. (2) 20 (1969), 129–137.CrossRefGoogle Scholar
Balasubramanian, R. and Shorey, T. N., ‘On the equation $a\left({x}^m-1\right)/ \left(x-1\right)=b\left({y}^n-1\right)/ \left(y-1\right)$ ’, Math. Scand. 46 (1980), 177–182.CrossRefGoogle Scholar
Bateman, P. T. and Stemmler, R. M., ‘Waring’s problem for algebraic number fields and primes of the form $({p}^r-1)/ ({p}^d-1)$ ’, Illinois J. Math. 6 (1962), 142–156.CrossRefGoogle Scholar
Bennett, M. A., Garbuz, B. and Marten, A., ‘Goormaghtigh’s equation: small parameters’, Publ. Math. Debrecen 96 (2020), 91–110.CrossRefGoogle Scholar
Bennett, M. A., Gherga, A. and Kreso, D., ‘An old and new approach to Goormaghtigh’s equation’, Trans. Amer. Math. Soc. 373 (2020), 5707–5745.CrossRefGoogle Scholar
Bugeaud, Y., Mignotte, M. and Siksek, S., ‘Classical and modular approaches to exponential Diophantine equations, I: Fibonacci and Lucas perfect powers’, Ann. of Math. (2) 163 (2006), 969–1018.CrossRefGoogle Scholar
Bugeaud, Y. and Shorey, T. N., ‘On the number of solutions of the generalized Ramanujan–Nagell equation’, J. reine angew. Math. 539 (2001), 55–74.Google Scholar
Bugeaud, Y. and Shorey, T. N., ‘On the Diophantine equation $\left({x}^m-1\right)/ \left(x-1\right)=\left({y}^n-1\right)/ \left(y-1\right)$ ’, Pacific J. Math. 207 (2002), 61–75.CrossRefGoogle Scholar
Davenport, H., Lewis, D. J. and Schinzel, A., ‘Equations of the form $f(x)=g(y)$ ’, Q. J. Math. Oxford Ser. (2) 12 (1961), 304–312.CrossRefGoogle Scholar
Dujella, A. and Pethő, A., ‘A generalization of a theorem of Baker and Davenport’, Q. J. Math. Oxford Ser. (2) 49 (1998), 291–306.CrossRefGoogle Scholar
He, B., ‘A remark on the Diophantine equation $({x}^3-1)/ (x-1)=({y}^n-1)/ (y-1)$ ’, Glas. Mat. Ser. III 44 (2009), 1–6.CrossRefGoogle Scholar
He, B. and Togbé, A., ‘On the number of solutions of Goormaghtigh equation for given $x$ and $y$ ’, Indag. Math. (N. S.) 19 (2008), 65–72.CrossRefGoogle Scholar
Karanikolov, K., ‘Sur une équation diophantienne considérée par Goormaghtigh’, Ann. Polon. Math. 14 (1963), 69–76 (in French).CrossRefGoogle Scholar
Laishram, S. and Shorey, T. N., ‘Baker’s explicit $abc$ -conjecture and applications’, Acta Arith. 155 (2012), 419–429.CrossRefGoogle Scholar
Le, M.-H., ‘On the Diophantine equation $({x}^3-1)/ (x-1)=({y}^n-1)/ (y-1)$ ’, Trans. Amer. Math. Soc. 351 (1999), 1063–1074.CrossRefGoogle Scholar
Le, M.-H., ‘Exceptional solutions of the exponential Diophantine equation $({x}^3-1)/ (x-1)=({y}^n-1)/ (y-1)$ ’, J. reine angew. Math. 543 (2002), 187–192.Google Scholar
Le, M.-H., ‘On Goormaghtigh’s equation $({x}^3-1)/ (x-1)=({y}^n-1)/ (y-1)$ ’, Acta Math. Sinica (Chinese Ser.) 45 (2002), 505–508 (in Chinese).Google Scholar
Leung, K.-H., Ma, S.-H. and Schmidt, B., ‘Proper partial geometries with Singer group and pseudogeometric partial difference sets’, J. Combin. Theory Ser. A 115 (2008), 147–177.CrossRefGoogle Scholar
Ljunggren, W., ‘Some theorems on indeterminate equations of the form $\left({x}^n-1\right)/ \left(x-1\right)={y}^q$ ’, Norsk Mat. Tidsskr. 25 (1943), 17–20 (in Norwegian).Google Scholar
Luca, F., ‘On an equation of Goormaghtigh’, Mosc. J. Comb. Number Theory 1 (2011), 74–88.Google Scholar
Makowski, A. and Schinzel, A., ‘Sur l’équation indéterminée de R. Goormaghtigh’, Mathesis 68 (1959), 128–142 (in French).Google Scholar
Makowski, A. and Schinzel, A., ‘Sur l’équation indéterminée de R. Goormaghtigh (Deuxième note)’, Mathesis 70 (1965), 94–96 (in French).Google Scholar
Matveev, E. M., ‘An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers II’, Izv. Math. 64 (2000), 1217–1269.CrossRefGoogle Scholar
Mignotte, M. and Voutier, P. M., ‘A kit for linear forms in three logarithms’, Preprint, 2023, arXiv:2205.08899v2; with an appendix by M. Laurent.CrossRefGoogle Scholar
Nesterenko, Y. V. and Shorey, T. N., ‘On the equation of Goormaghtigh’, Acta Arith. 83 (1998), 381–389.CrossRefGoogle Scholar
Ratat, R., Interméd. des math. 23 (1916), 150.Google Scholar
Rose, J. and Goormaghtigh, R., Interméd. des math. 24 (1917), 88–90.Google Scholar
Saradha, N., ‘Application of the explicit $abc$ -conjecture to two Diophantine equations’, Acta Arith. 151 (2012), 401–419.CrossRefGoogle Scholar
Shorey, T. N., ‘Integers with identical digits’, Acta Arith. 53 (1989), 187–205.CrossRefGoogle Scholar
Shorey, T. N., ‘Exponential Diophantine equations involving products of consecutive integers and related equations’, in: Number Theory (eds. Bambah, R. P., Dumir, V. C. and Hans-Gill, R. J.) (Birkhäuser, Basel, 2000), 463–495.Google Scholar
Shorey, T. N., ‘An equation of Goormaghtigh and Diophantine approximations’, in: Current Trends in Number Theory (eds. Adhikari, S. D., Katre, S. A. and Ramakrishnan, B.) (Hindustan Book Agency, New Delhi, 2002), 185–197.CrossRefGoogle Scholar
Shorey, T. N., ‘Diophantine approximations, Diophantine equations, transcendence and applications’, Indian J. Pure Appl. Math. 37 (2006), 9–39.Google Scholar
Shorey, T. N. and Tijdeman, R., ‘New applications of Diophantine approximations to Diophantine equations’, Math. Scand. 39 (1976), 5–18.CrossRefGoogle Scholar
Tijdeman, R., ‘Some applications of Diophantine approximation’, in: Number Theory for the Millennium III (eds. Bennett, M. A., Berndt, B. C., Boston, N., Diamond, H. G., Hildebrand, A. J. and Phillpp, W.) (A. K. Peters, Natick, MA, 2002), 261–284.Google Scholar
Yang, H. and Fu, R.-Q., ‘A note on the Goormaghtigh equation’, Period. Math. Hungar. 79 (2019), 86–93.CrossRefGoogle Scholar
Yuan, P.-Z., ‘On the Diophantine equation $({x}^3-1)/ (x-1)=({y}^n-1)/ (y-1)$ ’, J. Number Theory 112 (2005), 20–25.CrossRefGoogle Scholar