About some Diophantine equations with restrictions.
Acu, Dumitru (1999)
General Mathematics
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Acu, Dumitru (1999)
General Mathematics
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A. Schinzel (2015)
Colloquium Mathematicae
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All solutions of the equation x²+x+1 = yz in non-negative integers x,y,z are given in terms of an arithmetic continued fraction.
Ryuta Hashimoto (2001)
Acta Arithmetica
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Meignen, Pierrick (1999)
Beiträge zur Algebra und Geometrie
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Avraham Bourla (2014)
Acta Arithmetica
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We establish arithmetical properties and provide essential bounds for bi-sequences of approximation coefficients associated with the natural extension of maps, leading to continued fraction-like expansions. These maps are realized as the fractional part of Möbius transformations which carry the end points of the unit interval to zero and infinity, extending the classical regular and backwards continued fraction expansions.
H. NAKADA (1987-1988)
Seminaire de Théorie des Nombres de Bordeaux
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Yasushige Watase (2015)
Formalized Mathematics
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In this article we formalize some results of Diophantine approximation, i.e. the approximation of an irrational number by rationals. A typical example is finding an integer solution (x, y) of the inequality |xθ − y| ≤ 1/x, where 0 is a real number. First, we formalize some lemmas about continued fractions. Then we prove that the inequality has infinitely many solutions by continued fractions. Finally, we formalize Dirichlet’s proof (1842) of existence of the solution [12], [1]. ...
Kentaro Nakaishi (2006)
Acta Arithmetica
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Fritz Schweiger (2008)
Acta Arithmetica
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Boris Adamczewski (2010)
Acta Arithmetica
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Maosheng Xiong, Alexandru Zaharescu (2006)
Acta Arithmetica
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James Mc Laughlin (2008)
Acta Arithmetica
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Takao Komatsu (2003)
Acta Arithmetica
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Florin P. Boca, Joseph Vandehey (2012)
Acta Arithmetica
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Anton Lukyanenko, Joseph Vandehey (2015)
Acta Arithmetica
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We provide a generalization of continued fractions to the Heisenberg group. We prove an explicit estimate on the rate of convergence of the infinite continued fraction and several surprising analogs of classical formulas about continued fractions.