Murat Alan

Yildiz Technical University · Department of Mathematics

Let $ k \geq 2 $ and $ ( L_{n}^{(k)} )_{n \geq 2-k} $ be the $k-$generalized Lucas sequence with initial condition $ L_{2-k}^{(k)} = \cdots = L_{-1}^{(k)}=0 ,$ $ L_{0}^{(k,}=2,$ $ L_{1}^{(k)}=1$ and each term afterwards is the sum of the $ k $ preceding terms. A positive integer is an almost repdigit if its digits are all equal except for at most o...

In this paper, we define the notion of almost repdigit as a positive integer whose digits are all equal except for at most one digit, and we search all terms of the k-generalized Fibonacci sequence which are almost repdigits. In particular, we find all k-generalized Fibonacci numbers which are powers of 10 as a special case of almost repdigits. In...

Let (Pn)n≥0 be the Pell sequence defined by P0=0, P1=1 and Pn+2=2Pn+1+Pn for n≥0 and Mk=2k-1 be the k-th Mersenne number for k≥1. We show that the Diophantine equation PnPm=Mk with m≤n has only the unique positive integer solution (n,m,k)=(1,1,1).

Let $m$ be a positive integer. In this paper we consider the exponential Diophantine equation $(6m^{2}+1)^{x}+(3m^{2}-1)^{y}=(3m)^{z}$ and we show that it has only unique positive integer solution $(x,y,z)=(1,1,2)$ for all $ m>1. $ The proof depends on so called classification method and famous primitive divisor theorem.

Let $(F_n)_{n\geq 0}$ and $(L_n)_{n\geq 0}$ be the Fibonacci and Lucas sequences, respectively. In this paper we determine all Fibonacci numbers which are mixed concatenations of a Fibonacci and a Lucas numbers. By mixed concatenations of $ a $ and $ b $, we mean the both concatenations $\overline{ab}$ and $\overline{ba}$ together, where $ a $ and...

Let \((F_n)_{n\ge 0}\) and \((L_n)_{n\ge 0}\) be the Fibonacci and Lucas sequences. In this paper we determine all Fibonacci and Lucas numbers which are concatenations of two terms of the other sequence. This problem is identical to solve the Diophantine equations \( F_n=10^d L_m +L_k \) and \( L_n=10^d F_m+F_k \) in non-negative integers (n, m, k)...

Let m > 1 be a positive integer. We show that the exponential Diophantine equation mx + (m + 1)y = (1 + m + m2)z has only the positive integer solution (x, y, z) = (2, 1, 1) when m ≥ 2.

Let m≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m \ge 1$$\end{document} be a positive integer. We show that the exponential Diophantine equation (m2+m+1)x+my=(m+...

In this paper we find all positive integer solutions (x, y, n, a, b) of the equation in the title for non negative integers a and b under the condition that the integers x and y are relatively prime and \( n \ge 3\).

Let m be a positive integer. We show that the exponential Diophantine equation (18m²⁺¹)x+(7m²-1)y=(5m)z has only the positive integer solution (x,y,z) = (1,1,2) except for m ≡ 23,47,63,87 (mod 120). For m ≢ 0 (mod 5) we use some elementary methods and linear forms in two logarithms. For m ≡ 0 (mod 5) we apply a result for linear forms in p-adic log...

Let R be a commutative ring with identity and M be, not necessarily torsion-free, R-module. Unique factorization module (UFM) is introduced via U-decomposition and it is shown that M is a cyclic R-module is necessary but not sufficient condition for M to be a UFM.

In this study, we investigate the half-factorial property in non-maximal real quadratic orders and investigate some necessary and sufficient conditions for these orders to be half-factorial domains in terms of their conductor.

Let R be a commutative ring with identity. R is a finite factorization ring (FFR) if every nonzero nonunit of R has only a finite number of factorizations up to order and associates. In this article, we give a characterization of R for R[X] and R[[X]] to be an FFR.

We investigate the factorization properties on the polynomial extension [ ] A X of A where A is a UFR and show that [ ] A X is a U-BFR for any UFR A . We also consider the ring structure [ ] A XI X + where A is a UFR. A bir TÇA halka (tektürlü çarpanlara ayrılabilen halka) olmak üzere A 'nın polinom genişlemesi [ ] A X üzerindeki çarpanlara ayırma...

Some factorization properties are investigated in the context of commutative rings with zero divisors. Especially we introduce the notion of a finite decomposition ring (FDR) which is equivalent to an FFD for the domain case and by this definition each UFR is an FDR. Directed union of commutative rings having these factorization properties are also...