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Integer matrices related to Liouville's function

Shea-Ming Oon (2013)

Czechoslovak Mathematical Journal

In this note, we construct some integer matrices with determinant equal to certain summation form of Liouville's function. Hence, it offers a possible alternative way to explore the Prime Number Theorem by means of inequalities related to matrices, provided a better estimate on the relation between the determinant of a matrix and other information such as its eigenvalues is known. Besides, we also provide some comparisons on the estimate of the lower bound of the smallest singular value. Such discussion...

Integers with a maximal number of Fibonacci representations

Petra Kocábová, Zuzana Masáková, Edita Pelantová (2005)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

We study the properties of the function R ( n ) which determines the number of representations of an integer n as a sum of distinct Fibonacci numbers F k . We determine the maximum and mean values of R ( n ) for F k ≤ n < F k + 1 .

Integers with a maximal number of Fibonacci representations

Petra Kocábová, Zuzana Masáková, Edita Pelantová (2010)

RAIRO - Theoretical Informatics and Applications

We study the properties of the function R(n) which determines the number of representations of an integer n as a sum of distinct Fibonacci numbers Fk. We determine the maximum and mean values of R(n) for Fk ≤ n < Fk+1.

Introduction to Diophantine Approximation

Yasushige Watase (2015)

Formalized Mathematics

In this article we formalize some results of Diophantine approximation, i.e. the approximation of an irrational number by rationals. A typical example is finding an integer solution (x, y) of the inequality |xθ − y| ≤ 1/x, where 0 is a real number. First, we formalize some lemmas about continued fractions. Then we prove that the inequality has infinitely many solutions by continued fractions. Finally, we formalize Dirichlet’s proof (1842) of existence of the solution [12], [1].

Isomorphic digraphs from powers modulo p

Guixin Deng, Pingzhi Yuan (2011)

Czechoslovak Mathematical Journal

Let p be a prime. We assign to each positive number k a digraph G p k whose set of vertices is { 1 , 2 , ... , p - 1 } and there exists a directed edge from a vertex a to a vertex b if a k ≡ b ( mod p ) . In this paper we obtain a necessary and sufficient condition for G p k 1 ≃ G p k 2 .

Iterated digit sums, recursions and primality

Larry Ericksen (2006)

Acta Mathematica Universitatis Ostraviensis

We examine the congruences and iterate the digit sums of integer sequences. We generate recursive number sequences from triple and quintuple product identities. And we use second order recursions to determine the primality of special number systems.

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