On the Diophantine equation x^2+7^{alpha}.11^{beta}=y^n

In this paper, we give all the solutions of the Diophantine equation x^2+7^{alpha}.11^{beta}=y^n, in nonnegative integers x, y, n>=3 with x and y coprime, except for the case when alpha.x is odd and beta is even.Comment: to appear in Miskolc Mathematical Notes; MSC key words and phrases: Exponential equations, Primitive divisors of Lucas sequence

On the Diophantine equation ∑j=1kjFjp=Fnq\sum_{j=1}^kjF_j^p=F_n^q

Let $F_n$ denote the $n^{th}$ term of the Fibonacci sequence. In this paper, we investigate the Diophantine equation $F_1^p+2F_2^p+\cdots+kF_{k}^p=F_{n}^q$ in the positive integers $k$ and $n$, where $p$ and $q$ are given positive integers. A complete solution is given if the exponents are included in the set $\{1,2\}$. Based on the specific cases we could solve, and a computer search with $p,q,k\le100$ we conjecture that beside the trivial solutions only $F_8=F_1+2F_2+3F_3+4F_4$, $F_4^2=F_1+2F_2+3F_3$, and $F_4^3=F_1^3+2F_2^3+3F_3^3$ satisfy the title equation.Comment: 12 page

Rational Points on Elliptic Curves y^2=x^3+a^3 in f_{p} where p{\equiv}1(mod6) is Prime

In this work, we consider the rational points on elliptic curves over finite fields F_{p}. We give results concerning the number of points on the elliptic curve y^2{\equiv}x^3+a^3(mod p)where p is a prime congruent to 1 modulo 6. Also some results are given on the sum of abscissae of these points. We give the number of solutions to y^2{\equiv}x^3+a^3(modp), also given in ([1], p.174), this time by means of the quadratic residue character, in a different way, by using the cubic residue character. Using the Weil conjecture, one can generalize the results concerning the number of points in F_{p} to F_{p^{r}}.Comment: 9 pages; Keywords: Elliptic curves over finite fields, rational point

The development of a social network analysis software tool

Thesis (Master)--Izmir Institute of Technology, Computer Engineering, Izmir, 2009Includes bibliographical references (leaves: 33-34)Text in English; Abstract: Turkish and Englishix, 38 leavesNowadays the amount of spam is increasing in an uncontrollable way. This situation decreases the email trust of the Internet users and reduces the usability of the emails to the critical level. On the other hand, this makes a lot of money loss for the hosting companies. There are many anti-spam tools that are developed against spammers. Some of these tools are really succesful. However; since the spammers improve their techniques, spams gain immune to these tools. In this thesis, the problem of detecting spams in in-coming and out-going emails is adressed.. To achieve this goal, an email social network is constructed by using the traffic of emails between users.While constructing this network, only information that can be gained from the email structure is used. The results on the real data set show that the techniques applied have been effective and also point to new directions of research in this area