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Mathematical Proceedings of the Cambridge Philosophical Society Mathematical Proceedings of the Cambridge Philosophical Society

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On the digital representation of smooth numbers

Published online by Cambridge University Press:  29 August 2017

YANN BUGEAUD  and
HAJIME KANEKO
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
E-mail address:
bugeaud@math.unistra.fr
kanekoha@math.tsukuba.ac.jp
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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
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 165 , Issue 3 , November 2018 , pp. 533 - 540
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), 607–631.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), 525–540.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), 525–540.Google Scholar
[4] Blecksmith, R., Filaseta, M. and Nicol, C. A result on the digits of an. Acta Arith. 64 (1993), 331–339.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), 31–43.CrossRefGoogle Scholar
[7] Corvaja, P. and Zannier, U. S-unit points on analytic hypersurfaces. Ann. Sci. École Norm. Sup. 38 (2005), 76–92.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), 125–180 (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), 177–236.CrossRefGoogle Scholar
[10] Stewart, C. L. On the representation of an integer in two different bases. J. Reine Angew. Math. 319 (1980), 63–72.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), 257–264.Google Scholar
[12] Yu, K. p-adic logarithmic forms and group varieties. III. Forum Math. 19 (2007), 187–280.CrossRefGoogle Scholar

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