mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Prouhet–Thue–Morse constant, named for

Eugène Prouhet Eugene is a common male given name that comes from the Greek language, Greek εὐγενής (''eugenēs''), "noble", literally "well-born", from εὖ (''eu''), "well" and γένος (''genos''), "race, stock, kin".Axel Thue Axel Thue (; 19 February 1863 – 7 March 1922) was a Norwegian mathematician, known for his original work in diophantine approximation and combinatorics. Work Thue published his first important paper in 1909. He stated in 1914 the so-calle ...

, and

Marston Morse Harold Calvin Marston Morse (March 24, 1892 – June 22, 1977) was an American mathematician best known for his work on the ''calculus of variations in the large'', a subject where he introduced the technique of differential topology now known a ...

, is the number—denoted by —whose

binary expansion A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" ( one). The base-2 numeral system is a positional notatio ...

0.01101001100101101001011001101001... is given by the

Thue–Morse sequence In mathematics, the Thue–Morse sequence, or Prouhet–Thue–Morse sequence, is the binary sequence (an infinite sequence of 0s and 1s) obtained by starting with 0 and successively appending the Boolean complement of the sequence obtained thus ...

. That is, :

\tau = \sum_^ \frac = 0.412454033640 \ldots

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, :

\tau = \frac\left -\prod_^\left(1-\frac\right)\right

This formula is obtained by substituting ''x'' = 1/2 into generating series for :

F(x) = \sum_^ (-1)^ x^n = \prod_^ ( 1 - x^ )

The

continued fraction In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...

expansion of the constant is ; 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 Kurt Mahler FRS (26 July 1903, Krefeld, Germany – 25 February 1988, Canberra, Australia) was a German mathematician who worked in the fields of transcendental number theory, diophantine approximation, ''p''-adic analysis, and the geometry of ...

in 1929. He also showed that the number :

\sum_^ t_n \, \alpha^n

is also transcendental for any

algebraic number An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...

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 Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...

. If a

language Language is a structured system of communication. The structure of a language is its grammar and the free components are its vocabulary. Languages are the primary means by which humans communicate, and may be conveyed through a variety of met ...

''L'' over is chosen at random, by flipping a

fair coin In probability theory and statistics, a sequence of independent Bernoulli trials with probability 1/2 of success on each trial is metaphorically called a fair coin. One for which the probability is not 1/2 is called a biased or unfair coin. In the ...

to decide whether each word ''w'' is in ''L'', the probability that it contains at least one word for each possible length is :

p = \prod_^\left(1-\frac\right) = \sum_^ \frac = 2 - 4 \tau = 0.35018386544\ldots

Notes

References

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External links