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PERFECT POWERS WITH THREE DIGITS

Part of: Elementary number theory Diophantine equations Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  06 August 2013

Michael A. Bennett  and
Yann Bugeaud
Michael A. Bennett
Affiliation:
Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, V6T 1Z2,Canada email bennett@math.ubc.ca
Yann Bugeaud
Affiliation:
Mathématiques, Université de Strasbourg, 7, rue René Descartes, 67084 Strasbourg,France email bugeaud@math.unistra.fr
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Abstract

We solve the equation ${x}^{a} + {x}^{b} + 1= {y}^{q} $ in positive integers $x, y, a, b$ and $q$ with $a\gt b$ and $q\geq 2$ coprime to $\phi (x)$. This requires a combination of a variety of techniques from effective Diophantine approximation, including lower bounds for linear forms in complex and $p$-adic logarithms, the hypergeometric method of Thue and Siegel applied $p$-adically, local methods, and the algorithmic resolution of Thue equations.

MSC classification

Secondary: 11A63: Radix representation; digital problems 11D61: Exponential equations 11J86: Linear forms in logarithms; Baker's method
Type
Research Article
Information
Mathematika , Volume 60 , Issue 1 , January 2014 , pp. 66 - 84
Copyright
Copyright © University College London 2013 

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References

Bennett, M., Perfect powers with few ternary digits, Integers 12A (The John Selfridge Memorial Volume), 8pp., 2012.CrossRefGoogle Scholar
Bennett, M., Bugeaud, Y. and Mignotte, M., Perfect powers with few binary digits and related Diophantine problems. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) (to appear).Google Scholar
Bennett, M., Bugeaud, Y. and Mignotte, M., Perfect powers with few binary digits and related Diophantine problems, II. Math. Proc. Cambridge Philos. Soc. 153 (2012), 525–540.CrossRefGoogle Scholar
Beukers, F., On the generalized Ramanujan–Nagell equation II. Acta Arith. 39 (1981), 113–123.CrossRefGoogle Scholar
Bugeaud, Y., Linear forms in two $m$-adic logarithms and applications to Diophantine problems. Compositio Math. 132 (2002), 137–158.CrossRefGoogle Scholar
Bugeaud, Y., Mignotte, M. and Roy, Y., On the Diophantine equation $\frac{{x}^{n} - 1}{x- 1} = {y}^{q} $. Pacific J. Math. 193 (2000), 257–268.CrossRefGoogle Scholar
Bugeaud, Y., Mignotte, M., Roy, Y. and Shorey, T. N., The diophantine equation $({x}^{n} - 1)/ (x- 1)= {y}^{q} $ has no solution with $x$ square. Math. Proc. Cambridge Philos. Soc. 127 (1999), 353–372.Google Scholar
Chudnovsky, G. V., On the method of Thue–Siegel. Ann. of Math. (2) 117 (1983), 325–382.Google Scholar
Corvaja, P. and Zannier, U., On the Diophantine equation $f({a}^{m} , y)= {b}^{n} $. Acta Arith. 94 (2000), 25–40.CrossRefGoogle Scholar
Corvaja, P. and Zannier, U., Finiteness of odd perfect powers with four nonzero binary digits. Ann. Inst. Fourier (Grenoble) 63 (2013), 715–731.CrossRefGoogle Scholar
Le, M., A note on the diophantine equation $\frac{{x}^{m} - 1}{x- 1} = {y}^{n} $. Acta Arith. 64 (1993), 19–28.Google Scholar
Luca, F., The Diophantine equation ${x}^{2} = {p}^{a} \pm {p}^{b} + 1$. Acta Arith. 112 (2004), 87–101.CrossRefGoogle Scholar
Saradha, N. and Shorey, T. N., The equation $\frac{{x}^{n} - 1}{x- 1} = {y}^{q} $ with $x$ square. Math. Proc. Cambridge Philos. Soc. 125 (1999), 1–19.CrossRefGoogle Scholar
Scott, R., Elementary treatment of ${p}^{a} \pm {p}^{b} + 1= {x}^{2} $. Available online at the homepage of Robert Styer: http://www41.homepage.villanova.edu/robert.styer/ReeseScott/index.htm.Google Scholar
Szalay, L., The equations ${2}^{n} \pm {2}^{m} \pm {2}^{l} = {z}^{2} $. Indag. Math. (N.S.) 13 (2002), 131–142.CrossRefGoogle Scholar