Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-11T04:35:33.614Z Has data issue: false hasContentIssue false
  • English
  • Français

ON A PROBLEM ON NORMAL NUMBERS RAISED BY IGOR SHPARLINSKI

Part of: Multiplicative number theory Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  16 June 2011

JEAN-MARIE DE KONINCK  and
IMRE KÁTAI
JEAN-MARIE DE KONINCK*
Affiliation:
Dép. de mathématiques et de statistique, Université Laval, Québec, Québec G1V 0A6, Canada (email: jmdk@mat.ulaval.ca)
IMRE KÁTAI
Affiliation:
Computer Algebra Department, Eötvös Loránd University, 1117 Budapest, Pázmány Péter Sétány I/C, Hungary (email: katai@compalg.inf.elte.hu)
*
✉For correspondence; e-mail: jmdk@mat.ulaval.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Given an integer d≥2, a d-normal number, or simply a normal number, is an irrational number whosed-ary expansion is such that any preassigned sequence, of length k≥1, taken within this expansion occurs at the expected limiting frequency, namely 1/dk. Answering questions raised by Igor Shparlinski, we show that 0.P(2)P(3)P(4)…P(n)… and 0.P(2+1)P(3+1)P(5+1)…P(p+1)…, where P(n) stands for the largest prime factor of n, are both normal numbers.

Keywords

normal numbersprimesshifted primeslargest prime factor

MSC classification

Secondary: 11K16: Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. 11N37: Asymptotic results on arithmetic functions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 84 , Issue 2 , October 2011 , pp. 337 - 349
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[1]Bassily, N. L. and Kátai, I., ‘Distribution of consecutive digits in the q-ary expansions of some sequences of integers’, J. Math. Sci. 78(1) (1996), 11–17.CrossRefGoogle Scholar
[2]Borel, E., ‘Les probabilités dénombrables et leurs applications arithmétiques’, Rend. Circ. Mat. Palermo 27 (1909), 247–271.CrossRefGoogle Scholar
[3]Champernowne, D. G., ‘The construction of decimals normal in the scale of ten’, J. Lond. Math. Soc. 8 (1933), 254–260.CrossRefGoogle Scholar
[4]Copeland, A. H. and Erdős, P., ‘Note on normal numbers’, Bull. Amer. Math. Soc. 52 (1946), 857–860.CrossRefGoogle Scholar
[5]Davenport, H. and Erdős, P., ‘Note on normal decimals’, Canad. J. Math. 4 (1952), 58–63.CrossRefGoogle Scholar
[6]De Koninck, J. M. and Kátai, I., ‘Construction of normal numbers by classified prime divisors of integers’, Funct. Approx., to appear.Google Scholar
[7]De Koninck, J. M. and Kátai, I., ‘On the distribution of subsets of primes in the prime factorization of integers’, Acta Arith. 72(2) (1995), 169–200.CrossRefGoogle Scholar
[8]Halberstam, H. H. and Richert, H. E., Sieve Methods (Academic Press, London, 1974).Google Scholar
[9]Madritsch, M. G., Thuswaldner, J. M. and Tichy, R. F., ‘Normality of numbers generated by the values of entire functions’, J. Number Theory 128 (2008), 1127–1145.CrossRefGoogle Scholar
[10]Nakai, Y. and Shiokawa, I., ‘Normality of numbers generated by the values of polynomials at primes’, Acta Arith. 81(4) (1997), 345–356.CrossRefGoogle Scholar
[11]Tenenbaum, G., Introduction à la théorie analytique des nombres (Collection Échelles, Belin, 2008).Google Scholar