Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-29T19:15:25.721Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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.Google 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.Google 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.Google Scholar
[7] Corvaja, P. and Zannier, U. S-unit points on analytic hypersurfaces. Ann. Sci. École Norm. Sup. 38 (2005), 76–92.Google 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.Google 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.Google Scholar