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Digit patterns and transcendental numbers

Part of: Elementary number theory

Published online by Cambridge University Press:  09 April 2009

Patrick Morton  and
W. J. Mourant
Patrick Morton
Affiliation:
Wellesley CollegeWellesley, Massachusetts 02181, U.S.A.
W. J. Mourant
Affiliation:
37 William J. Heights Framingham, Massachusetts 01701, U.S.A.
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Abstract

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We use a theorem of Loxton and van der Poorten to prove the transcendence of certain real numbers defined by digit patterns. Among the results we prove are the following. If k is an integer at least 2, P is any nonzero pattern of digits base k, and counts the number of occurrences (mod r) of p in the base k representation of n, then is transcendental except when k = 3, P = 1 and r = 2. When (r, k − 1) = 1 the linear span of the numbers has infinite dimension over Q, where P ranges over all patterns base k without leading zeros.

MSC classification

Secondary: 11A63: Radix representation; digital problems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 51 , Issue 2 , October 1991 , pp. 216 - 236
Copyright
Copyright © Australian Mathematical Society 1991

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