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ON THE NUMBER OF SOLUTIONS OF THE DIOPHANTINE EQUATION axm−byn=c

Part of: Diophantine equations

Published online by Cambridge University Press:  13 January 2010

BO HE  and
ALAIN TOGBÉ
BO HE
Affiliation:
Department of Mathematics, ABA Teachers College, Wenchuan, Sichuan 623000, PR China (email: [email protected])
ALAIN TOGBÉ*
Affiliation:
Mathematics Department, Purdue University North Central, 1401 South US 421, Westville IN 46391, USA (email: [email protected], [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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Let a, b, c, x and y be positive integers. In this paper we sharpen a result of Le by showing that the Diophantine equation has at most two positive integer solutions (m,n) satisfying min (m,n)>1.

Keywords

exponential equationsDiophantine equations

MSC classification

Secondary: 11D61: Exponential equations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 81 , Issue 2 , April 2010 , pp. 177 - 185
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

Footnotes

The first author was supported by the Applied Basic Research Foundation of Sichuan Provincial Science and Technology Department (No. 2009JY0091). The second author is grateful to Purdue University North Central for the support.

References

[1]Bennett, M., ‘On some exponential equations of S. S. Pillai’, Canad. J. Math. 53 (2001), 897–922.CrossRefGoogle Scholar
[2]Bugeaud, Y. and Luca, F., ‘On Pillai’s diophantine equation’, New York J. Math. 12 (2006), 193–217.Google Scholar
[3]Bugeaud, Y. and Shorey, T. N., ‘On the diophantine equation ’, Pacific J. Math. 207 (2002), 61–75.CrossRefGoogle Scholar
[4]Cao, Z. F., ‘On the equation ax m−by n=2’, Kexue Tongbao 35 (1990), 558–559 (in Chinese).Google Scholar
[5]Cassels, J. W. S., ‘On the equation a x−b y=1. II’, Proc. Cambridge Philos. Soc. 56 (1960), 97–103.CrossRefGoogle Scholar
[6]Dickson, L. E., History of the Theory of Numbers, Vol. II (Carnegie Institution of Washington. Reprinted by Chelsea Publ. Co., New York, 1971).Google Scholar
[7]He, B. and Togbé, A., ‘On the number of solutions of Goormaghtigh equation for given x and y’, Indag. Math. (N.S.) 19(1) (2008), 65–72.CrossRefGoogle Scholar
[8]Herschfeld, A., ‘The equation 2x−3y=d’, Bull. Amer. Math. Soc. 42 (1936), 231–234.CrossRefGoogle Scholar
[9]Le, M., ‘A note on the diophantine equation ax m−by n=k’, Indag. Math. 3(2) (1992), 185–191.Google Scholar
[10]LeVeque, W. J., ‘On the equation a x−b y=1’, Amer. J. Math. 74 (1952), 325–331.CrossRefGoogle Scholar
[11]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
[12]Mihailescu, P., ‘Primary cyclotomic units and a proof of Catalan’s conjecture’, J. Reine Angew. Math. 572 (2004), 167–195.Google Scholar
[13] PARI/GP, version 2.1.7, Bordeaux, 2005, http://pari.math.u-bordeaux.fr/.Google Scholar
[14]Pillai, S. S., ‘On the inequality 0<a x−b y≤n’, J. Indian Math. Soc. 19 (1931), 1–11.Google Scholar
[15]Pillai, S. S., ‘On a x−b y=c’, J. Indian Math. Soc. (N.S.) 2 (1936), 119–122.Google Scholar
[16]Ribenboim, P., Catalan’s Conjecture (Academic Press, London, 1994).Google Scholar
[17]Ribenboim, P., My Numbers, My Friends: Popular Lectures on Number Theory (Springer, Berlin, 2000).CrossRefGoogle Scholar
[18]Shorey, T. N., ‘On the equation ax m−by n=k’, Indag. Math. 48 (1986), 353–358.CrossRefGoogle Scholar
[19]Stroeker, R. J. and Tijdeman, R., Diophantine equations, Computational Methods in Number Theory, Part II, Math. Centre Tracts, 155 (Math. Centrum, Amsterdam, 1982), pp. 321–369.Google Scholar