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A note on some discrete valuation rings of arithmetical functions

Emil Daniel Schwab, Gheorghe Silberberg (2000)

Archivum Mathematicum

The paper studies the structure of the ring A of arithmetical functions, where the multiplication is defined as the Dirichlet convolution. It is proven that A itself is not a discrete valuation ring, but a certain extension of it is constructed,this extension being a discrete valuation ring. Finally, the metric structure of the ring A is examined.

A note on the congruence n p k m p k ≡ n m ( mod p r )

Romeo Meštrović (2012)

Czechoslovak Mathematical Journal

In the paper we discuss the following type congruences: n p k m p k ≡ m n ( mod p r ) , where p is a prime, n , m , k and r are various positive integers with n ≥ m ≥ 1 , k ≥ 1 and r ≥ 1 . Given positive integers k and r , denote by W ( k , r ) the set of all primes p such that the above congruence holds for every pair of integers n ≥ m ≥ 1 . Using Ljunggren’s and Jacobsthal’s type congruences, we establish several characterizations of sets W ( k , r ) and inclusion relations between them for various values k and r . In particular, we prove that W ( k + i , r ) = W ( k - 1 , r ) for all k ≥ 2 , i ≥ 0 and 3 ≤ r ≤ 3 k , and W ( k , r ) = W ( 1 , r ) for...

A note on univoque self-sturmian numbers

Jean-Paul Allouche (2008)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

We compare two sets of (infinite) binary sequences whose suffixes satisfy extremal conditions: one occurs when studying iterations of unimodal continuous maps from the unit interval into itself, but it also characterizes univoque real numbers; the other is a disguised version of the set of characteristic sturmian sequences. As a corollary to our study we obtain that a real number β in ( 1 , 2 ) is univoque and self-sturmian if and only if the β -expansion of 1 is of the form 1 v , where v is a characteristic...

A note on univoque self-Sturmian numbers

Jean-Paul Allouche (2010)

RAIRO - Theoretical Informatics and Applications

We compare two sets of (infinite) binary sequences whose suffixes satisfy extremal conditions: one occurs when studying iterations of unimodal continuous maps from the unit interval into itself, but it also characterizes univoque real numbers; the other is a disguised version of the set of characteristic Sturmian sequences. As a corollary to our study we obtain that a real number β in (1,2) is univoque and self-Sturmian if and only if the β-expansion of 1 is of the form 1v, where v is a characteristic...

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