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Prouhet–Thue–Morse constant
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In [[mathematics]], the '''Prouhet–Thue–Morse constant''', named for {{ill|Eugène Prouhet|fr}}, [[Axel Thue]], and [[Marston Morse]], is the number—denoted by {{mvar|τ}}—whose [[binary expansion]] 0.01101001100101101001011001101001... is given by the [[Prouhet–Thue–Morse sequence]]. That is, :<math> \tau = \sum_{n=0}^{\infty} \frac{t_n}{2^{n+1}} = 0.412454033640 \ldots </math> where {{math|''t<sub>n</sub>''}} is the {{math|''n''<sup>th</sup>}} element of the Prouhet–Thue–Morse sequence. ==Other representations== The Prouhet–Thue–Morse constant can also be expressed, without using {{math|''t<sub>n</sub>''}} , as an infinite product,<ref name="mw">{{Mathworld|Thue-MorseConstant|Thue-Morse Constant}}</ref> :<math> \tau = \frac{1}{4}\left[2-\prod_{n=0}^{\infty}\left(1-\frac{1}{2^{2^n}}\right)\right] </math> This formula is obtained by substituting ''x'' = 1/2 into generating series for {{math|''t<sub>n</sub>''}} :<math> F(x) = \sum_{n=0}^{\infty} (-1)^{t_n} x^n = \prod_{n=0}^{\infty} ( 1 - x^{2^n} ) </math> The [[simple continued fraction|continued fraction expansion]] of the constant is [0; 2, 2, 2, 1, 4, 3, 5, 2, 1, 4, 2, 1, 5, 44, 1, 4, 1, 2, 4, 1, …] {{OEIS|A014572}} Yann Bugeaud and Martine Queffélec showed that infinitely many partial quotients of this continued fraction are 4 or 5, and infinitely many partial quotients are greater than or equal to 50.<ref>{{cite journal |last1=Bugeaud |first1=Yann |last2=Queffélec |first2=Martine |title=On Rational Approximation of the Binary Thue-Morse-Mahler Number |journal=Journal of Integer Sequences |date=2013 |volume=16 |issue=13.2.3 |url=https://cs.uwaterloo.ca/journals/JIS/VOL16/Bugeaud/bugeaud3.html}}</ref> ==Transcendence== The Prouhet–Thue–Morse constant was shown to be [[transcendental number|transcendental]] by [[Kurt Mahler]] in 1929.<ref>{{cite journal | first=Kurt | last=Mahler | authorlink=Kurt Mahler | title=Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen | journal=[[Math. Annalen]] | volume=101 | year=1929 | pages=342–366 | jfm=55.0115.01 | doi=10.1007/bf01454845| s2cid=120549929 }}</ref> He also showed that the number :<math>\sum_{i=0}^{\infty} t_n \, \alpha^n</math> is also transcendental for any [[algebraic number]] α, where 0 < |''α''| < 1. Yann Bugaeud proved that the Prouhet–Thue–Morse constant has an [[irrationality measure]] of 2.<ref>{{cite journal |last1=Bugaeud |first1=Yann |title=On the rational approximation to the Thue–Morse–Mahler numbers |journal=Annales de l'Institut Fourier |date=2011 |volume=61 |issue=5 |pages=2065–2076 |doi=10.5802/aif.2666 |url=https://aif.centre-mersenne.org/item/AIF_2011__61_5_2065_0/|doi-access=free }}</ref> ==Appearances== The Prouhet–Thue–Morse constant appears in [[probability]]. If a [[Formal language|language]] ''L'' over {0, 1} is chosen at random, by flipping a [[fair coin]] to decide whether each word ''w'' is in ''L'', the probability that it contains at least one word for each possible length is <ref>{{cite journal |last1=Allouche |first1=Jean-Paul |last2=Shallit |first2=Jeffrey |title=The Ubiquitous Prouhet–Thue–Morse Sequence |journal=Discrete Mathematics and Theoretical Computer Science |date=1999 |page=11 |url=http://www.cs.uwaterloo.ca/~shallit/Papers/ubiq.ps}}</ref> :<math> p = \prod_{n=0}^{\infty}\left(1-\frac{1}{2^{2^n}}\right) = \sum_{n=0}^{\infty} \frac{(-1)^{t_n}}{2^{n+1}} = 2 - 4 \tau = 0.35018386544\ldots</math> ==See also== * [[Euler–Mascheroni constant]] * [[Fibonacci word]] * [[Golay–Rudin–Shapiro sequence]] * [[Komornik–Loreti constant]] ==Notes== <references /> ==References== *{{cite book | last1 = Allouche | first1 = Jean-Paul | last2 = Shallit | first2 = Jeffrey | author2-link = Jeffrey Shallit | isbn = 978-0-521-82332-6 | publisher = [[Cambridge University Press]] | title = Automatic Sequences: Theory, Applications, Generalizations | year = 2003 | zbl=1086.11015 }}. * {{cite book | last=Pytheas Fogg | first=N. | editor1=Berthé, Valérie|editor1-link=Valérie Berthé|editor2=Ferenczi, Sébastien|editor3=Mauduit, Christian|editor4=Siegel, Anne | title=Substitutions in dynamics, arithmetics and combinatorics | series=Lecture Notes in Mathematics | volume=1794 | location=Berlin | publisher=[[Springer-Verlag]] | year=2002 | isbn=3-540-44141-7 | zbl=1014.11015 }} ==External links== * {{OEIS el|sequencenumber=A010060|name=Thue–Morse sequence|formalname=Thue–Morse sequence: let A_k denote the first 2^k terms; then A_0 = 0 and for k >= 0, A_{k+1} = A_k B_k, where B_k is obtained from A_k by interchanging 0's and 1's}} * [https://www.cs.uwaterloo.ca/~shallit/Papers/ubiq.ps The ubiquitous Prouhet–Thue–Morse sequence], John-Paull Allouche and Jeffrey Shallit, (undated, 2004 or earlier) provides many applications and some history * [https://planetmath.org/prouhetthuemorseconstant PlanetMath entry] {{DEFAULTSORT:Prouhet-Thue-Morse Constant}} [[Category:Mathematical constants]] [[Category:Number theory]] [[Category:Real transcendental numbers]] {{Numtheory-stub}}
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