In mathematics, the Prouhet–Thue–Morse constant, named for Eugène Prouhet, Axel Thue, and Marston Morse, is the number—denoted by —whose binary expansion 0.01101001100101101001011001101001... is given by the Thue–Morse sequence. That is, where is the element of the Prouhet–Thue–Morse sequence. Other representations The Prouhet–Thue–Morse constant can also be expressed, without using , as an infinite product, This formula is obtained by substituting x = 1/2 into generating series for The 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, …] 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. Transcendence The Prouhet–Thue–Morse constant was shown to be transcendental by Kurt Mahler in 1929. He also showed that the number 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. Appearances The Prouhet–Thue–Morse constant appears in probability. If a 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 See also Euler-Mascheroni constant Fibonacci word Golay–Rudin–Shapiro sequence Komornik–Loreti constant Notes References . External links The ubiquitous Prouhet-Thue-Morse sequence, John-Paull Allouche and Jeffrey Shallit, (undated, 2004 or earlier) provides many applications and some history PlanetMath entry Mathematical constants Number theory Real transcendental numbers