mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Prouhet–Thue–Morse constant, named for ,

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-called w ...

, and Marston Morse, 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 for representing numbers that uses only two symbols for the natural numbers: typically "0" ( zero) and "1" ( one). A ''binary number'' may als ...

0.01101001100101101001011001101001... is given by the Prouhet–Thue–Morse sequence. 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 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 in 1929. He also showed that the number :

\sum_^ t_n \, \alpha^n

is also transcendental for any

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

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 a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...

. If a

language Language is a structured system of communication that consists of grammar and vocabulary. It is the primary means by which humans convey meaning, both in spoken and signed language, signed forms, and may also be conveyed through writing syste ...

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

fair coin In probability theory and statistics, a sequence of Independence (probability theory), 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 ca ...

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