On a problem on normal numbers raised by Igor Shparlinski

Authors

  • Jean-Marie De Koninck
  • Imre Katai

Keywords:

normal numbers, shifted primes, largest prime factor

Abstract

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

Published

2011-10-19

Issue

Vol. 84 No. 2 (2011)

Section

Articles
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