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On the digital representation of smooth numbers

Published online by Cambridge University Press:  29 August 2017

YANN BUGEAUD
Affiliation:
Institut de Recherche Mathématique Avancée, U.M.R. 7501, Université de Strasbourg et C.N.R.S., 7, rue René Descartes, 67084 Strasbourg, France. e-mail: bugeaud@math.unistra.fr
HAJIME KANEKO
Affiliation:
Institute of Mathematics, University of Tsukuba, 1-1-1, Tennodai, Tsukuba, Ibaraki, 305-8571, Japan. Center for Integrated Research in Fundamental Science and Engineering (CiRfSE), University of Tsukuba, 1-1-1, Tennodai, Tsukuba, Ibaraki, 305-8571, Japan. e-mail: kanekoha@math.tsukuba.ac.jp
Corresponding

Abstract

Let b ⩾ 2 be an integer. Among other results we establish, in a quantitative form, that any sufficiently large integer which is not a multiple of b cannot simultaneously be divisible only by very small primes and have very few nonzero digits in its representation in base b.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2017 

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References

[1] Barat, G., Tichy, R. F. and Tijdeman, R. Digital blocks in linear numeration systems. In: Number Theory in Progress, vol. 2 (Zakopane–Kościelisko, 1997), (de Gruyter, Berlin, 1999), 607631.Google Scholar
[2] Bennett, M. A., Bugeaud, Y. and Mignotte, M. Perfect powers with few binary digits and related Diophantine problems. II. Math. Proc. Camb. Phil. Soc. 153 (2012), 525540.CrossRefGoogle Scholar
[3] Bennett, M. A., Bugeaud, Y. and Mignotte, M. Perfect powers with few binary digits and related Diophantine problems. Ann. Sc. Norm. Super. Pisa Cl. Sci. 12 (2013), 525540.Google Scholar
[4] Blecksmith, R., Filaseta, M. and Nicol, C. A result on the digits of an. Acta Arith. 64 (1993), 331339.CrossRefGoogle Scholar
[5] Bugeaud, Y. On the digital representation of integers with bounded prime factors. Osaka J. Math. To appear.Google Scholar
[6] Bugeaud, Y., Cipu, M. and Mignotte, M. On the representation of Fibonacci and Lucas numbers in an integer base. Ann. Math. Qué. 37 (2013), 3143.CrossRefGoogle Scholar
[7] Corvaja, P. and Zannier, U. S-unit points on analytic hypersurfaces. Ann. Sci. École Norm. Sup. 38 (2005), 7692.CrossRefGoogle Scholar
[8] Matveev, E. M. An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers. II. Izv. Ross. Acad. Nauk Ser. Mat. 64 (2000), 125180 (in Russian); English translation in Izv. Math. 64 (2000), 1217–1269.Google Scholar
[9] Schinzel, A. On two theorems of Gelfond and some of their applications. Acta Arith. 13 (1967), 177236.CrossRefGoogle Scholar
[10] Stewart, C. L. On the representation of an integer in two different bases. J. Reine Angew. Math. 319 (1980), 6372.Google Scholar
[11] Stewart, C. L. On the greatest square-free factor of terms of a linear recurrence sequence. In: Diophantine Equations. Tata Inst. Fund. Res. Stud. Math. 20 (Tata Inst. Fund. Res., Mumbai, 2008), 257264.Google Scholar
[12] Yu, K. p-adic logarithmic forms and group varieties. III. Forum Math. 19 (2007), 187280.CrossRefGoogle Scholar

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