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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: [email protected])
IMRE KÁTAI
Affiliation:
Computer Algebra Department, Eötvös Loránd University, 1117 Budapest, Pázmány Péter Sétány I/C, Hungary (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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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

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