On Hurwitzian and Tasoev's continued fractions
On infinite series representations of real numbers
On integers n with for .
On integers not of the form n - φ (n)
W. Sierpiński asked in 1959 (see [4], pp. 200-201, cf. [2]) whether there exist infinitely many positive integers not of the form n - φ(n), where φ is the Euler function. We answer this question in the affirmative by proving Theorem. None of the numbers (k = 1, 2,...) is of the form n - φ(n).
On iteration digraph and zero-divisor graph of the ring
In the first part, we assign to each positive integer a digraph whose set of vertices consists of elements of the ring with the addition and the multiplication operations modulo and for which there is a directed edge from to if and only if . Associated with are two disjoint subdigraphs: and whose union is The vertices of are coprime to and the vertices of are not coprime to In this part, we study the structure of in detail. In the second part, we investigate the zero-divisor...
On Jacobsthal's g(n)-Function.
On -imperfect numbers.
On Krasner's Criteria for the First Case of Fermat's Last Theorem.
On K-Stage Euclidean algorithms for Galois extensions of Q.
On k-triad sequences.
On linear normal lattices configurations
In this paper we extend Champernowne’s construction of normal numbers in base to the case and obtain an explicit construction of the generic point of the shift transformation of the set . We prove that the intersection of the considered lattice configuration with an arbitrary line is a normal sequence in base .
On Lucas pseudoprimes of the form in arithmetic progression with a prescribed value of the Jacobi symbol
On m times covering systems of congruences
On mod logarithms and .
On multiplicatively dependent linear numeration systems, and periodic points
Two linear numeration systems, with characteristic polynomial equal to the minimal polynomial of two Pisot numbers and respectively, such that and are multiplicatively dependent, are considered. It is shown that the conversion between one system and the other one is computable by a finite automaton. We also define a sequence of integers which is equal to the number of periodic points of a sofic dynamical system associated with some Parry number.
On multiplicatively dependent linear numeration systems, and periodic points
Two linear numeration systems, with characteristic polynomial equal to the minimal polynomial of two Pisot numbers β and γ respectively, such that β and γ are multiplicatively dependent, are considered. It is shown that the conversion between one system and the other one is computable by a finite automaton. We also define a sequence of integers which is equal to the number of periodic points of a sofic dynamical system associated with some Parry number.
On multiplicatively e-perfect numbers.
On multiplicatively perfect numbers.
On and .
On near-perfect and deficient-perfect numbers
For a positive integer n, let σ(n) denote the sum of the positive divisors of n. Let d be a proper divisor of n. We call n a near-perfect number if σ(n) = 2n + d, and a deficient-perfect number if σ(n) = 2n - d. We show that there is no odd near-perfect number with three distinct prime divisors and determine all deficient-perfect numbers with at most two distinct prime factors.