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ON A PROBLEM OF ERDÖS AND MAHLER CONCERNING CONTINUED FRACTIONS

Part of: Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  19 October 2016

JEAN LELIS  and
DIEGO MARQUES
JEAN LELIS
Affiliation:
Departamento de Matemática, Universidade de Brasília, Brasília, 70910-900, Brazil email [email protected]
DIEGO MARQUES*
Affiliation:
Departamento de Matemática, Universidade de Brasília, Brasília, 70910-900, Brazil email [email protected]
*
✉[email protected]
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Abstract

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In 1939, Erdös and Mahler [‘Some arithmetical properties of the convergents of a continued fraction’, J. Lond. Math. Soc. (2)14 (1939), 12–18] studied some arithmetical properties of the convergents of a continued fraction. In particular, they raised a conjecture related to continued fractions and Liouville numbers. In this paper, we shall apply the theory of linear forms in logarithms to obtain a result in the direction of this problem.

Keywords

continued fractionsLiouville numberslinear forms in logarithms

MSC classification

Primary: 11J70: Continued fractions and generalizations
Secondary: 11J86: Linear forms in logarithms; Baker's method
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 95 , Issue 2 , April 2017 , pp. 183 - 186
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

References

Bugeaud, Y., Approximation by Algebraic Numbers, Cambridge Tracts in Mathematics, 160 (Cambridge University Press, New York, 2004).CrossRefGoogle Scholar
Erdös, P. and Mahler, K., ‘Some arithmetical properties of the convergents of a continued fraction’, J. Lond. Math. Soc. (2) 14 (1939), 12–18.Google Scholar
Fraenkel, A. S., ‘On a theorem of D. Ridout in the theory of Diophantine approximations’, Trans. Amer. Math. Soc. 105 (1962), 84–101.Google Scholar
Fraenkel, A. S., ‘Transcendental numbers and a conjecture of Erdös and Mahler’, J. Lond. Math. Soc. (2) 39 (1964), 405–416.Google Scholar
Mahler, K., ‘Ein Analog zu einem Schneiderschen Satz, I’, Proc. Kon. Ned. Akad. Wetensch. 39 (1936), 633–640.Google Scholar
Mahler, K., ‘Ein Analog zu einem Schneiderschen Satz, II’, Proc. Kon. Ned. Akad. Wetensch. 39 (1936), 729–737.Google Scholar
Matveev, E. M., ‘An explict lower bound for a homogeneous rational linear form in logarithms of algebraic numbers, II’, Izv. Ross. Akad. Nauk Ser. Mat. 64 (2000), 125–180; English transl. Izv. Math. 64 (2000), 1217–1269.Google Scholar
Ridout, D., ‘Rational approximations to algebraic numbers’, Mathematika 4 (1957), 125–131.CrossRefGoogle Scholar
Shorey, T. N., ‘Divisors of convergents of a continued fraction’, J. Number Theory 17 (1983), 127–133.CrossRefGoogle Scholar