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PERFECT POWERS WITH THREE DIGITS
Part of:
Diophantine approximation, transcendental number theory
Elementary number theory
Diophantine equations
Published online by Cambridge University Press: 06 August 2013
Abstract
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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
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- Copyright © University College London 2013
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
$m$-adic logarithms and applications to Diophantine problems. Compositio Math. 132 (2002), 137–158.CrossRefGoogle ScholarBugeaud, 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
$\frac{{x}^{n} - 1}{x- 1} = {y}^{q} $. Pacific J. Math. 193 (2000), 257–268.CrossRefGoogle ScholarBugeaud, 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
$({x}^{n} - 1)/ (x- 1)= {y}^{q} $ has no solution with 
$x$ square. Math. Proc. Cambridge Philos. Soc. 127 (1999), 353–372.Google ScholarChudnovsky, 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
$f({a}^{m} , y)= {b}^{n} $. Acta Arith. 94 (2000), 25–40.CrossRefGoogle ScholarCorvaja, 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
$\frac{{x}^{m} - 1}{x- 1} = {y}^{n} $. Acta Arith. 64 (1993), 19–28.Google ScholarLuca, F., The Diophantine equation ${x}^{2} = {p}^{a} \pm {p}^{b} + 1$. Acta Arith. 112 (2004), 87–101.CrossRefGoogle Scholar
${x}^{2} = {p}^{a} \pm {p}^{b} + 1$. Acta Arith. 112 (2004), 87–101.CrossRefGoogle ScholarSaradha, 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
$\frac{{x}^{n} - 1}{x- 1} = {y}^{q} $ with 
$x$ square. Math. Proc. Cambridge Philos. Soc. 125 (1999), 1–19.CrossRefGoogle ScholarScott, 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
${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 ScholarSzalay, L., The equations ${2}^{n} \pm {2}^{m} \pm {2}^{l} = {z}^{2} $. Indag. Math. (N.S.) 13 (2002), 131–142.CrossRefGoogle Scholar
${2}^{n} \pm {2}^{m} \pm {2}^{l} = {z}^{2} $. Indag. Math. (N.S.) 13 (2002), 131–142.CrossRefGoogle Scholar
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