Hostname: page-component-8448b6f56d-dnltx Total loading time: 0 Render date: 2024-04-24T19:59:35.126Z Has data issue: false hasContentIssue false
  • English
  • Français

ERDŐS’ METHOD FOR DETERMINING THE IRRATIONALITY OF PRODUCTS

Part of: Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  13 July 2011

JAROSLAV HANČL  and
ONDŘEJ KOLOUCH
JAROSLAV HANČL*
Affiliation:
Department of Mathematics and Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, 30. dubna 22, 701 03 Ostrava 1, Czech Republic (email: hancl@osu.cz)
ONDŘEJ KOLOUCH
Affiliation:
Department of Mathematics and Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, 30. dubna 22, 701 03 Ostrava 1, Czech Republic (email: ondrej.kolouch@osu.cz)
*
✉For correspondence; e-mail: hancl@osu.cz
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper deals with a sufficient condition for the infinite product of rational numbers to be an irrational number. The condition requires only some conditions for convergence and does not use other properties like divisibility. The proof is based on an idea of Erdős.

Keywords

irrationalityinfinite products

MSC classification

Secondary: 11J72: Irrationality; linear independence over a field
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 84 , Issue 3 , December 2011 , pp. 414 - 424
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

The first author was supported by the grants no. ME09017 and MSM 6198898701.

References

[1]Badea, C., ‘The irrationality of certain infinite products’, Studia Univ. Babeş-Bolyai Math. 31(3) (1986), 3–8.Google Scholar
[2]Badea, C., ‘The irrationality of certain infinite series’, Glasg. Math. J. 29(2) (1987), 221–228.Google Scholar
[3]Duverney, D., ‘Sur les series de nombres rationnels a convergence rapide’, C. R. Acad. Sci., Paris, Ser. I, Math. 328(7) (1999), 553–556.CrossRefGoogle Scholar
[4]Erdős, P., ‘Some problems and results on the irrationality of the sum of infinite series’, J. Math. Sci. 10 (1975), 1–7.Google Scholar
[5]Erdős, P., ‘Erdős problem no. 6’, in: 1995 Prague Midsummer Combinatorial Workshop, KAM Series (95–309) (ed. Klazar, M.) (KAM MPP UK, Prague, 1995), p. 5.Google Scholar
[6]Genčev, M., ‘Infinite product for e 6ζ(3)’, Ann. Math. Sil. 21 (2007), 41–48, 2008.Google Scholar
[7]Genčev, M., ‘Evaluation of infinite series involving special products and their algebraic characterization’, Math. Slovaca 59(3) (2009), 365–378.Google Scholar
[8]Hančl, J., Nair, R., Sobková, S. and Šustek, J., ‘On expressible sets and p-adic numbers’, Proc. Edinb. Math. Soc. (2), to appear.Google Scholar
[9]Hančl, J., Rucki, P. and Šustek, J., ‘A generalization of Sándor’s theorem using iterated logarithms’, Kumamoto J. Math. 19 (2006), 25–36.Google Scholar
[10]Hančl, J. and Sobková, S., ‘Special linearly unrelated sequences’, J. Math. Kyoto Univ. 46(1) (2006), 31–45.Google Scholar
[11]Hančl, J., Štěpnička, J. and Šustek, J., ‘Linearly unrelated sequences and Erdős’ problem’, Ramanujan J. 17(3) (2008), 331–342.Google Scholar
[12]Laohakosol, V. and Kuhapatanakul, K., ‘The reverse irrationality criteria of Brun and Badea’, East-West J. Math. Special vol. (2008), 217–234.Google Scholar
[13]Laohakosol, V. and Kuhapatanakul, K., ‘The irrationality criteria of Brun and Badea are essentially equivalent’, in: Diophantine Analysis and Related Fields—DARF 2007/2008, AIP Conference Proceedings, 976 (American Institute of Physics, Melville, NY, 2008), pp. 144–159.Google Scholar
[14]Laohakosol, V. and Kuhapatanakul, K., ‘Irrationality criteria for infinite products’, J. Comb. Number Theory 1(1) (2008), 49–57.Google Scholar
[15]Zhou, P., ‘On the irrationality of a certain multivariable infinite product’, Quaest. Math. 29(3) (2006), 351–365.Google Scholar
[16]Zhou, P. and Lubinsky, D. S., ‘On the irrationality of ∏ ∞j=0(1±q −jr+q −2js)’, Analysis (Munich) 17(2–3) (1997), 129–153.Google Scholar