Related papers: Some Diophantine Equations involving associated Pe…
In this paper, we find all integers $c$ having at least two representations as a difference between linear recurrent sequences. This problem is a pillai problem involving Padovan and Fibonacci sequence. The proof of our main theorem uses…
In this paper, we find all repdigits which can be expressed as the product of a Narayana, and a product of two repdigits is Narayana.
Let $b\ge 2$ be a given integer. In this paper, we show that there only finitely many positive integers $d$ which are not squares, such that the Pell equation $X^2-dY^2=1$ has two positive integer solutions $(X,Y)$ with the property that…
The aim of this paper is to introduce Bell polynomials and numbers of the second kind and poly-Bell polynomials and numbers of the second kind, and to derive their explicit expressions, recurrence relations and some identities involving…
We observe that the computation $5^2 = 25$ has the digital property of the result being equal to the exponent concatenated directly to the left of the base. The generalization to a Diophantine equation and inequality in number bases has…
Let $ \{P_n\}_{n\geq 0} $ be the sequence of Perrin numbers defined by $P_0=3$, $P_1=0$,$P_2=2$ and $P_{n+3}=P_{n+1}+P_{n}$ for all $n \geq 0$. In this paper, we determine all Perrin numbers that are palindromic concatenations of two…
In this paper, dual complex Pell numbers and quaternions are defined. Also, some algebraic properties of dual-complex Pell numbers and quaternions which are connected with dual complex numbers and Pell numbers are investigated. Furthermore,…
Let $\beta$ be a non-unit real algebraic integer greater than one and $\{a_{n}\}_{n \geq 0}$ be a sequence satisfying a linear recurrence relation $a_{n+3}=aa_{n+2}+ba_{n+1}+ca_{n}$. Under certain conditions, we prove that the number of…
In this paper, we find all the solutions of the Diophantine equation $P_\ell + P_m +P_n=2^a$, in nonnegative integer variables $(n,m,\ell, a)$ where $P_k$ is the $k$-th term of the Pell sequence $\{P_n\}_{n\ge 0}$ given by $P_0=0$, $P_1=1$…
Let $g\ge 2$ be an integer and $\mathcal R_g\subset \mathbb N$ be the set of repdigits in base $g$. Let $\mathcal D_g$ be the set of Diophantine triples with values in $\mathcal R_g$; that is, $\mathcal D_g$ is the set of all triples…
This thesis is about the study of Diophantine equations involving binary recurrent sequences with arithmetic functions. Various Diophantine problems are investigated and new results are found out of this study. Firstly, we study several…
For positive integers $b\geq 2$, $k<b$, and $t$, we say that an integer $k_b^{(t)}$ is a $b$-repdigit if $k_b^{(t)}$ can be expressed as the digit $k$ repeated $t$ times in base-$b$ representation, i.e., $k_b^{(t)} =k(b^t-1)/(b-1)$. In the…
In this paper, bicomplex Pell and bicomplex Pell-Lucas numbers are defined. Also, negabicomplex Pell and negabicomplex Pell-Lucas numbers are given. Some algebraic properties of bicomplex Pell and bicomplex Pell-Lucas numbers which are…
Let $ \{L_n\}_{n\geq 0} $ be the sequence of Lucas numbers. In this paper, we determine all Lucas numbers that are palindromic concatenations of two distinct repdigits.
In this paper, we construct Pell matrices, analogous to Fibonacci matrices, to study algebraic properties of Pell numbers via linear algebra. This framework yields identities involving the trace, inverse, and determinant, as well as matrix…
We investigate cases where Thabit and Williams numbers in base $b$ can be expressed as the sum or difference of two $g$-repdigits. For specific values of $b$ and $g$, we describe parametric solutions yielding infinitely many solutions for…
By means of two complex-valued functions (depending on an integer parameter P>=1) we construct helices of integer ratio R>=1 related to the so-called Binet formulae for P-Lucas and P-Fibonacci sequences. Based on these functions a new map…
We introduce new refinements of the Bell, factorial, and unsigned Stirling numbers of the first and second kind that unite the derangement, involution, associated factorial, associated Bell, incomplete Stirling, restricted factorial,…
In this extended abstract we deal with the relations between the numerical/diophantine approximation and the symbolic/algebraic geometry approachs to solving of multivariate diophentine polynomial systems, obtaining several consecuences…
We reduce the principal problem of Additive Number Theory of whether an infinite sequence of integers constitutes a finite basis for the integers to a Diophantine problem involving the difference set of the sequence, by proving a formula…