arXiv subjects
Publications and source records attributed to Diana Savin.
{\small In this paper we define and we study properties of} $\left(l,1,p+2q,q\cdot l\right) -$ {\small numbers,} $\left(l,1,p+2q,q\cdot l\right) -$ {\small quaternions,} $\left(l,1,p+2q,q\cdot l\right) -$ {\small symbol elements. Finally, we obtain an algebraic structure with these elements.}
In this paper we define and study properties and applications of a, b, x0, x1 elements in some special cases.
In this paper we present applications of some special numbers obtained from a difference equation of degree three, especially in the Coding Theory. As a particular case, we obtain the generalized Pell-Fibonacci-Lucas numbers, which were extended to the generalized quaternions.
In this paper we determine sufficient conditions for a quaternion algebra to split over a quadratic field. In the last section of the paper, we find a class of division symbol algebras of degree $n$ (where $n$ is a positive integer, $n\geq 3$) over a $p-$ adic field or over a cyclotomic field.
In this paper we study special Fibonacci quaternions and special generalized Fibonacci-Lucas quaternions in quaternion algebras over finite fields.
In this paper we determine some properties of Fibonacci octonions. Also, we introduce the generalized Fibonacci-Lucas octonions and we investigate some properties of these elements.
In this paper, we introduce the generalized Fibonacci-Lucas quaternions and we prove that the set of these elements is an order,in the sense of ring theory, of a quaternion algebra. Moreover, we investigate some properties of these elements.
{\small In this paper, we find a class of division quaternion algebras over the field }$\mathbb{Q}\left( i\right) ${\small \ and a class of division symbol algebras over a cyclotomic field.}
In this paper we study certain quaternion algebras and symbol algebras which split.
In this paper we will study some properties of the matrix representations of symbol algebras of degree three, we study some equations with coefficients in these algebras, we find an octonion algebra in a symbol algebra of degree three, we define the Fibonacci symbol elements and we give some properties of them.
In this paper we provide an algorithm to compute the product between two elements in a symbol algebra of degree n and we find an octonion non-division algebra in a symbol algebra of degree three. Starting from this last idea, we try to find an answer to the question if there are division symbol algebras of degree three. The answer is positive and we provide, using MAGMA software, some examples of division symbol algebras of degree 3 and of degree 5. Moreover, we will give some interesting applications of the symbol algebras in number theory.
In this paper, we prove a theorem about the integer solutions to the Diophantine equation $x^{4}-q^{4}=py^{r}$, extending previous work of K.Gy\H ory, and F.Luca and A.Togbe, and of the author.
In this paper we study the Diophantine equation $x^{4}-q^{4}=py^{5},$ with the following conditions: $p$ and $q$ are different prime natural numbers, $y$ is not divisible with $p$, $p\equiv3$ (mod20), $q\equiv4$ (mod5), $\overline{p}$ is a generator of the group $(U(\textbf{Z}_{q^{4}}),\cdot)$, $(x,y)=1$, 2 is a 5-power residue mod $q$.
We give a necessary condition for the existence of solutions of the Diophantine equation $p=x^{q}+ry^{q},$ with $p$, $q$, $r$ distinct odd prime natural numbers.
In this paper, we obtain some properties of the symbol algebras, starting from their connections with the quaternion and cyclic algebras over a field $K_{p},$% where $K$ is an algebraic number field, $p$ is a prime in $K$ and $K_{p}$ is the completion of $K$ with respect to $p-$ adic valuation, in the case when $K=\Bbb{Q}(\varepsilon), \varepsilon ^{3}=1,\varepsilon \neq 1.$