Copeland, Arthur H.; Erdős, Paul Note on normal numbers. (English) Zbl 0063.00962 Bull. Am. Math. Soc. 52, 857-860 (1946). From the text: D. G. Champernowne [J. Lond. Math. Soc. 8, 254–260 (1933; Zbl 0007.33701, JFM 59.0214.01)] proved that the infinite decimal \(0.123456789101112\cdots\) was normal (in the sense of Borel) with respect to the base 10, a normal number being one whose digits exhibit a complete randomness.More precisely a number is normal provided each of the digits \(0, 1, 2, \cdots, 9\) occurs with a limiting relative frequency of \(1/10\) and each of the \(10^k\) sequences of \(k\) digits occurs with the frequency \(10^{-k}\). Champernowne conjectured that if the sequence of all integers were replaced by the sequence of primes then the corresponding decimal \(0.12357111317\cdots\) would be normal with respect to the base 10. We propose to show not only the truth of his conjecture but to obtain a somewhat more general result, namely:Theorem. If \(a_1, a_2, \dots,\) is an increasing sequence of integers such that for every \(\theta < 1\) the number of \(a\)’s up to \(N\) exceeds \(N^\theta\) provided \(N\) is sufficiently large, then the infinite decimal \(0.a_1a_2a_3\cdots\) is normal with respect to the base \(\beta\) in which these integers are expressed.On the basis of this theorem the conjecture of Champernowne follows from the fact that the number of primes up to \(N\) exceeds \(cN'/\log N\) for any \(c < 1\) provided \(N\) is sufficiently large. The corresponding result holds for the sequence of integers which can be represented as the sum of two squares since every prime of the form \(4k+1\) is also of the form \(x^2+2\) and the number of these primes up to \(N\) exceeds \(c'N/\log N\) for sufficiently large \(N\) when \(c'<1/2\).The above theorem is based on the concept of \((\varepsilon, k)\)-normality of A. S. Besicovitch [Math. Z. 39, 146–156 (1934; Zbl 0009.20002, JFM 60.0937.01)]. Cited in 4 ReviewsCited in 57 Documents MSC: 11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. Citations:Zbl 0007.33701; JFM 59.0214.01; Zbl 0009.20002; JFM 60.0937.01 PDFBibTeX XMLCite \textit{A. H. Copeland} and \textit{P. Erdős}, Bull. Am. Math. Soc. 52, 857--860 (1946; Zbl 0063.00962) Full Text: DOI Online Encyclopedia of Integer Sequences: Decimal expansion of Champernowne constant (or Mahler’s number), formed by concatenating the positive integers. Decimal expansion of Copeland-Erdős constant: concatenate primes. Decimal expansion of the binary Champernowne constant 0.862240125868... whose binary expansion is the concatenation of 1, 2, 3, ... written in binary. Decimal expansion of the ”binary” Copeland-Erdős constant 0.734121515408286120606...: concatenate primes in base two. Decimal expansion of constant equal to concatenated semiprimes. Decimal expansion of constant equal to concatenated nonprimes. Decimal expansion of the constant formed by concatenating the imperfect numbers. Write the semiprimes backwards in base 10 and juxtapose (concatenate) their digits. Square array read by descending antidiagonals: The n-th row gives the decimal expansion of the base-n Champernowne constant.