The algebraic independence of certain Liouville continued fractions
HTML articles powered by AMS MathViewer
- by William W. Adams PDF
- Proc. Amer. Math. Soc. 95 (1985), 512-516 Request permission
Abstract:
This work uses some simple Liouville type arguments to extend some recent work of Bundschuh and of Laohakosol and Ubolsri on algebraic independence. The results are stronger and are not restricted to just two numbers. We then use the results to give a new and simple proof of Bundschuh’s result concerning the algebraic independence of certain numbers whose $g$-adic and continued fraction expansions are both known.References
- William W. Adams, On the algebraic independence of certain Liouville numbers, J. Pure Appl. Algebra 13 (1978), no. 1, 41–47. MR 508729, DOI 10.1016/0022-4049(78)90041-5
- William W. Adams and J. L. Davison, A remarkable class of continued fractions, Proc. Amer. Math. Soc. 65 (1977), no. 2, 194–198. MR 441879, DOI 10.1090/S0002-9939-1977-0441879-4
- P. E. Böhmer, Über die Transzendenz gewisser dyadischer Brüche, Math. Ann. 96 (1927), no. 1, 367–377 (German). MR 1512324, DOI 10.1007/BF01209172
- Peter Bundschuh, Über eine Klasse reeller transzendenter Zahlen mit explizit angebbarer $g$-adischer und Kettenbruch-Entwicklung, J. Reine Angew. Math. 318 (1980), 110–119 (German). MR 579386, DOI 10.1515/crll.1980.318.110
- Peter Bundschuh, Transcendental continued fractions, J. Number Theory 18 (1984), no. 1, 91–98. MR 734440, DOI 10.1016/0022-314X(84)90045-3
- L. V. Danilov, Certain classes of transcendental numbers, Mat. Zametki 12 (1972), 149–154 (Russian). MR 316391
- J. L. Davison, A series and its associated continued fraction, Proc. Amer. Math. Soc. 63 (1977), no. 1, 29–32. MR 429778, DOI 10.1090/S0002-9939-1977-0429778-5
- Alain Durand, Indépendance algébrique de nombres complexes et critère de transcendance, Compositio Math. 35 (1977), no. 3, 259–267 (French). MR 457364
- Yuval Z. Flicker, Algebraic independence by a method of Mahler, J. Austral. Math. Soc. Ser. A 27 (1979), no. 2, 173–188. MR 531112
- Serge Lang, Introduction to diophantine approximations, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966. MR 0209227
- Vichian Laohakosol and Patchara Ubolsri, Some algebraically independent continued fractions, Proc. Amer. Math. Soc. 95 (1985), no. 2, 169–173. MR 801317, DOI 10.1090/S0002-9939-1985-0801317-0
- William Judson LeVeque, Topics in number theory. Vols. 1 and 2, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1956. MR 0080682 R. Pass, Results concerning the algebraic independence of sets of Liouville numbers, Ph. D. Thesis, Univ. of Maryland, College Park, Md., 1978.
- Iekata Shiokawa, Algebraic independence of certain gap series, Arch. Math. (Basel) 38 (1982), no. 5, 438–442. MR 666917, DOI 10.1007/BF01304813
Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 95 (1985), 512-516
- MSC: Primary 11J85; Secondary 11J70
- DOI: https://doi.org/10.1090/S0002-9939-1985-0810154-2
- MathSciNet review: 810154