De Koninck, Jean-Marie; Kátai, Imre On a problem on normal numbers raised by Igor Shparlinski. (English) Zbl 1231.11086 Bull. Aust. Math. Soc. 84, No. 2, 337-349 (2011). For positive integers \(d\) and \(n\) with \(d\geq2\) we denote by \[ n=\varepsilon_0(n)+\varepsilon_1(n)d+\cdots+\varepsilon_t(n)d^t \] the \(d\)-ary representation with \(\varepsilon_i(n)\in E:=\{0,1,\ldots,d-1\}\) for \(0\leq i\leq t\) and \(\varepsilon_t(n)\neq0\). To this representation we associate the word \(\overline{n}=\varepsilon_0(n)\varepsilon_1(n)\ldots\varepsilon_t(n)\in E^{t+1}\). Furthermore for a real \(x\in[0,1)\) we denote by \[ x=0.a_1a_2a_3\ldots \] with \(a_i\in E\) the \(d\)-ary expansion of \(x\). We call a real \(x\in[0,1)\) normal if for any positive integer \(k\) and any block \(B\in E^k\) of digits of length \(k\) the number of occurrences of this block within the \(d\)-ary expansion is equal to the expected limiting frequency, namely \(d^{-k}\).The authors of the present answer questions raised by Igor Shparlinski in connection with the construction of normal numbers by describing their \(d\)-ary expansion. Let \(f\in\mathbb Z[X]\) be a polynomial with positive leading coefficient and \(P(n)\) denote the largest prime factor of \(n\). Then they are able to show that \[ 0.\overline{f(P(2))}\,\overline{f(P(3))}\,\overline{f(P(4))}\ldots\overline{f(P(n))}\ldots \] and \[ 0.\overline{f(P(2+1))}\,\overline{f(P(3+1))}\,\overline{f(P(5+1))}\ldots\overline{f(P(p+1))}\ldots, \] where \(p\) runs over the primes, are normal numbers. Reviewer: Manfred G. Madritsch (Graz) Cited in 1 ReviewCited in 4 Documents MSC: 11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. 11A41 Primes 11N37 Asymptotic results on arithmetic functions Keywords:normal numbers; primes; shifted primes; largest prime factor PDFBibTeX XMLCite \textit{J.-M. De Koninck} and \textit{I. Kátai}, Bull. Aust. Math. Soc. 84, No. 2, 337--349 (2011; Zbl 1231.11086) Full Text: DOI References: [1] DOI: 10.1016/j.jnt.2007.04.005 · Zbl 1213.11151 [2] Halberstam, Sieve Methods (1974) [3] De Koninck, Acta Arith. 72 pp 169– (1995) [4] DOI: 10.4153/CJM-1952-005-3 · Zbl 0046.04902 [5] Nakai, Acta Arith. 81 pp 345– (1997) [6] DOI: 10.1112/jlms/s1-8.4.254 · Zbl 0007.33701 [7] DOI: 10.1007/BF03019651 · JFM 40.0283.01 [8] DOI: 10.1007/BF02367950 [9] Tenenbaum, Introduction à la théorie analytique des nombres (2008) [10] DOI: 10.1090/S0002-9904-1946-08657-7 · Zbl 0063.00962 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.