Revista Matemática Iberoamericana

Arithmetic properties of positive integers with fixed digit sum

Florian Luca

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In this paper, we look at various arithmetic properties of the set of those positive integers $n$ whose sum of digits in a fixed base $b>1$ is a fixed positive integers $s$. For example, we prove that such integers can have many prime factors, that they are not very smooth, and that most such integers have a large prime factor dividing the value of their Euler $\phi$ function.

Article information

Rev. Mat. Iberoamericana, Volume 22, Number 2 (2006), 369-412.

First available in Project Euclid: 26 October 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11A63: Radix representation; digital problems {For metric results, see 11K16}
Secondary: 11N64: Other results on the distribution of values or the characterization of arithmetic functions

sum of digits smooth numbers subspace theorem linear forms in logarithms


Luca, Florian. Arithmetic properties of positive integers with fixed digit sum. Rev. Mat. Iberoamericana 22 (2006), no. 2, 369--412.

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  • Baker, A. and Wüstholz, G.: Logarithmic forms and group varieties. J. Reine Angew. Math. 442 (1993), 19-62.
  • Balog, A. and Wooley, T.: On strings of consecutive integers with no large prime factors. J. Austral Math. Soc. Ser. A 64 (1998), 266-276.
  • Banks, W. and Shparlinski, I. E.: Arithmetic properties of numbers with restricted digits. Acta Arith. 112 (2004), 313-332.
  • Bugeaud, Y.: Lower bounds for the greatest prime factor of $ax^m+by^n$. Acta. Math. Inform. Univ. Ostraviensis 6 (1998), 53-57.
  • Carmichael, R. D.: On the numerical factors of the arithmetic forms $\alpha^n\pm \beta^n$. Ann. of Math. (2) 15 (1913), 30-70.
  • Corvaja, P. and Zannier, U.: Diophantine equations with power sums and universal Hilbert sets. Indag. Math. (N.S.) 9 (1998), 317-332.
  • Dartyge, C. and Mauduit, C.: Nombres presque premiers dont l'écriture en base $r$ ne comporte pas certaines chiffres. J. Number Theory 81 (2000), 270-291.
  • De Bruijn, N. G.: On the number of positive integers $\le x$ and free of prime factors $>y$. Nederl. Akad. Wetensch. Proc. Ser. A 54 (1951), 50-60.
  • Eggleton, R. B. and Selfridge, J. L.: Consecutive integers with no large prime factors. J. Austral. Math. Soc. Ser. A 22 (1976), 1-11.
  • Fouvry, E. and Mauduit, C.: Methódes de crible et fonctions sommes des chiffres. Acta Arith. 77 (1996), 339-351.
  • Fouvry, E. and Mauduit, C.: Sommes des chiffres et nombres presque premiers. Math. Ann. 305 (1996), 571-599.
  • Konyagin, S., Mauduit, C. and Sárközy, A.: On the number of prime factors of integers characterized by digit properties. Period. Math. Hungar. 40 (2000), 37-52.
  • Luca, F.: The number of non zero digits of $n!$. Canad. Math. Bull. 45 (2002), 115-118.
  • Luca, F.: How smooth is $\phi(2^n+3)$? Rocky Mountain J. Math. 34 (2004), 1367-1389.
  • Mauduit, C. and Sárközy, A.: On the arithmetic structure of sets characterized by sum of digits properties. J. Number Theory 61 (1996), 25-38.
  • Mauduit, C. and Sárközy, A.: On the arithmetic structure of the integers whose sum of digits is fixed. Acta Arith. 81 (1997), 145-173.
  • Nicolas, J.-L.: Petites valeurs de la fonction d'Euler. J. Number Theory 17 (1983), 375-388.
  • Rosser, J. B. and Schoenfeld, L.: Approximate formulas for some functions of prime numbers. Illinois J. Math. 6 (1962) 64-94.
  • Schlickewei, H. P.: $S$-unit equations over number fields. Invent. Math. 102 (1990), no. 1, 95-107.
  • Schmidt, W. M.: Diophantine approximations and Diophantine equations. Lecture Notes in Mathematics 1467. Springer-Verlag, Berlin, 1991.
  • Senge, H. G. and Strauss, E. G.: $PV$-numbers and sets of multiplicity. Period. Math. Hungar. 3 (1973), 93-100.
  • Shparlinski, I. E.: Prime divisors of sparse integers. Period. Math. Hungar. 46 (2003), no. 2, 215-222.
  • Stewart, C. L.: On the representation of an integer in two different bases. J. Reine Angew. Math. 319 (1980), 63-72.
  • Szalay, L.: The equations $2^n\pm 2^m\pm 2^l=z^2$. Indag. Math. (N.S.) 13 (2002), 131-142.
  • Yu, K.: $p$-adic logarithmic forms and group varieties. II. Acta Arith. 89 (1999), no. 4, 337-378.