Osaka Journal of Mathematics

On the digital representation of integers with bounded prime factors

Yann Bugeaud

Full-text: Open access

Abstract

Let $b \ge 2$ be an integer. Not much is known on the representation in base $b$ of prime numbers or of numbers whose prime factors belong to a given, finite set. Among other results, we establish that any sufficiently large integer which is not a multiple of $b$ and has only small (in a suitable sense) prime factors has at least four nonzero digits in its representation in base $b$.

Article information

Source
Osaka J. Math., Volume 55, Number 2 (2018), 315-324.

Dates
First available in Project Euclid: 18 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1524038730

Mathematical Reviews number (MathSciNet)
MR3787747

Zentralblatt MATH identifier
06870391

Subjects
Primary: 11A63: Radix representation; digital problems {For metric results, see 11K16} 11J86: Linear forms in logarithms; Baker's method 11J87: Schmidt Subspace Theorem and applications

Citation

Bugeaud, Yann. On the digital representation of integers with bounded prime factors. Osaka J. Math. 55 (2018), no. 2, 315--324. https://projecteuclid.org/euclid.ojm/1524038730


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References

  • M.A. Bennett: Perfect powers with few ternary digits, Integers 12 (Selfridge Memorial Volume) (2012), 1159–1166.
  • M.A. Bennett and Y. Bugeaud: Perfect powers with three digits, Mathematika 60 (2014), 66–84.
  • M.A. Bennett, Y. Bugeaud and M. Mignotte: Perfect powers with few binary digits and related Diophantine problems, II, Math. Proc. Cambridge Philos. Soc. 153 (2012), 525–540.
  • M.A. Bennett, Y. Bugeaud and M. Mignotte: Perfect powers with few binary digits and related Diophantine problems, Ann. Sc. Norm. Super. Pisa Cl. Sci. 12 (2013), 941–953.
  • M.A. Bennett, M. Filaseta and O. Trifonov: Yet another generalization of the Ramanujan-Nagell equation, Acta Arith. 134 (2008), 211–217.
  • M.A. Bennett, M. Filaseta and O. Trifonov: On the factorization of consecutive integers, J. Reine Angew. Math. 629 (2009), 171–200.
  • J. Bourgain: Estimates on exponential sums related to Diffie–Hellman distributions, Geom. Funct. Anal. 15 (2005), 1–34.
  • J. Bourgain: Prescribing the binary digits of primes, II, Israel J. Math. 206 (2015), 165–182.
  • Y. Bugeaud and J.-H. Evertse: $S$-parts of terms of integer linear recurrence sequences, Mathematika 63 (2017), 840–851.
  • Y. Bugeaud, J.-H. Evertse and K. Györy: $S$-parts of values of univariate polynomials, binary forms and decomposable forms at integral points, to appear in Acta Arith.
  • P. Corvaja and U. Zannier: On the Diophantine equation $f(a^m,y)=b^n$, Acta Arith. 94 (2000), 25–40.
  • P. Corvaja and U. Zannier: $S$-unit points on analytic hypersurfaces, Ann. Sci. École Norm. Sup. (4) 38 (2005), 76–92.
  • P. Corvaja and U. Zannier: Finiteness of odd perfect powers with four nonzero binary digits, Ann. Inst. Fourier (Grenoble) 63 (2013), 715–731.
  • C. Elsholtz: Almost all primes have a multiple of small Hamming weight, Bull. Aust. Math. Soc. 94 (2016), 224–235.
  • S.S. Gross and A.F. Vincent: {On the factorization of $f(n)$ for $f(x)$ in ${\mathbf Z}[x]$, Int. J. Number Theory 9 (2013), 1225–1236.
  • K. Györy and K. Yu: Bounds for the solutions of $S$-unit equations and decomposable form equations, Acta Arith. 123 (2006), 9–41.
  • E.M. Matveev: An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers. II, Izv. Ross. Akad. Nauk Ser. Mat. 64 (2000), 125–180 (in Russian); English translation in Izv. Math. 64 (2000), 1217–1269.
  • Ch. Mauduit and J. Rivat: Sur un problème de Gelfond : la somme des chiffres des nombres premiers, Ann. of Math. 171 (2010), 1591–1646.
  • J. Maynard: Primes with restricted digits, Preprint, Available at: http://arxiv.org/pdf/1604.01041v1.pdf
  • W.M. Schmidt: Simultaneous approximations to algebraic numbers by rationals, Acta Math. 125 (1970), 189–201.
  • W.M. Schmidt: Norm form equations, Ann. of Math. 96 (1972), 526–551.
  • W.M. Schmidt: Diophantine Approximation, Lecture Notes in Math. 785, Springer, Berlin, 1980.
  • I. Shparlinski: Prime divisors of sparse integers, Period. Math. Hungar. 46 (2003), 215–222.
  • I. Shparlinski: Exponential sums and prime divisors of sparse integers, Period. Math. Hungar. 57 (2008), 93–99.
  • C.L. Stewart: On the representation of an integer in two different bases, J. Reine Angew. Math. 319 (1980), 63–72.
  • C.L. Stewart: On the greatest square-free factor of terms of a linear recurrence sequence, In: Diophantine equations, 257–264, Tata Inst. Fund. Res. Stud. Math. 20, Tata Inst. Fund. Res., Mumbai, 2008.
  • C.L. Stewart: On prime factors of terms of linear recurrence sequences, In: Number theory and related fields, 341–359, Springer Proc. Math. Stat. 43, Springer, New York, 2013.
  • L. Szalay: The equations $2^n \pm 2^m \pm 2^l=z^2$, Indag. Math. (N.S.) 13 (2002), 131–142.
  • K. Yu: $p$-adic logarithmic forms and group varieties. III, Forum Math. 19 (2007), 187–280.