Functiones et Approximatio Commentarii Mathematici

Construction of normal numbers by classified prime divisors of integers

Jean-Marie De Koninck and Imre Kátai

Full-text: Open access


Given an integer $d\ge 2$, a $d$-{\it normal number}, or simply a {\it normal number}, is a real number whose $d$-ary expansion is such that any preassigned sequence, of length $k\ge 1$, of base $d$ digits from this expansion, occurs at the expected frequency, namely $1/d^k$. We construct large families of normal numbers using classified prime divisors of integers.

Article information

Funct. Approx. Comment. Math., Volume 45, Number 2 (2011), 231-253.

First available in Project Euclid: 12 December 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11K16: Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. [See also 11A63]
Secondary: 11N37: Asymptotic results on arithmetic functions 11A41: Primes

normal numbers primes shifted primes arithmetic function


De Koninck, Jean-Marie; Kátai, Imre. Construction of normal numbers by classified prime divisors of integers. Funct. Approx. Comment. Math. 45 (2011), no. 2, 231--253. doi:10.7169/facm/1323705815.

Export citation


  • D.G.,Champernowne, The construction of decimals normal in the scale of ten, J. London Math. Soc. 8 (1933), 254-260.
  • A.H. Copeland and P. Erdős, Note on normal numbers, Bull. Amer. Math. Soc. 52 (1946), 857-860.
  • H.,Davenport and P.,Erdős, Note on normal decimals, Can. J. Math. 4 (1952), 58-63.
  • J.M. De Koninck and I. Kátai, On the distribution of subsets of primes in the prime factorization of integers, Acta Arithmetica 72(2) (1995), 169-200.
  • P.D.T.A.,Elliott, Probabilistic Number Theory I, Mean Value Theorems, Springer-Verlag, Berlin, 1979
  • H.H.,Halberstam and H.E.,Richert, Sieve Methods, Academic Press, London, 1974.
  • G.H.Hardy and S.Ramanujan, The normal number of prime factors of a number $n$, Quart.J.Math. 48 (1917), 76-92.
  • M.G.,Madritsch, J.M.,Thuswaldner and R.F.,Tichy, Normality of numbers generated by the values of entire functions, J. of Number Theory 128 (2008), 1127-1145.
  • Y.,Nakai and I.,Shiokawa, Normality of numbers generated by the values of polynomials at primes, Acta Arith. 81(4) (1997), 345-356.
  • G.,Tenenbaum, Introduction à la théorie analytique des nombres, Collection Échelles, Belin, 2008.