Prouhet–Thue–Morse constant

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In mathematics, the Prouhet–Thue–Morse constant, named for Eugène Prouhet, Axel Thue, and Marston Morse, is the number—denoted by —whose binary expansion .01101001100101101001011001101001... is given by the Thue–Morse sequence. That is,

where is the ith element of the Prouhet–Thue–Morse sequence.

The generating series for the is given by

and can be expressed as

This is the product of Frobenius polynomials, and thus generalizes to arbitrary fields.

The Prouhet–Thue–Morse constant was shown to be transcendental by Kurt Mahler in 1929.[1]


  1. ^ Mahler, Kurt (1929). "Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen". Math. Annalen. 101: 342–366. doi:10.1007/bf01454845. JFM 55.0115.01.

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