arXiv subjects
Publications and source records attributed to Bilge Peker.
A set of m distinct positive integers {a_{1},...a_{m}} is called a Diophantine m-tuple if a_{i}a_{j}+n is a square for each 1\leqi<j\leqm . The aim of this study is to show that some P_{k} sets can not be extendible to a Diophantine quadruple when k=2 and k=-3 and also to give some properties about P_{k} sets.
In this study, we find continued fraction expansion of sqrt(d) when d=a^2+2a where a is positive integer. We consider the integer solutions of the Pell equation x^2-(a^2+2a)y^2=N when N={-1,+1,-4,+4}. We formulate the n-th solution (x_{n},y_{n}) by using the continued fraction expansion. We also formulate the n-th solution (x_{n},y_{n}) via the generalized Fibonacci and Lucas sequences.
In this study, we find continued fraction expansion of sqrt(d) when d=a^2b^2-b and d=a^2b^2-2b where a and b are positive integers. We consider the integer solutions of the Pell equations x^2-(a^2b^2-b)y^2=N and x^2-(a^2b^2-2b)y^2=N when N is {+-1,+-4}. We formulate the n-th solution (x_{n},y_{n}) by using the continued fraction expansion. We also formulate the n-th solution (x_{n},y_{n}) in terms of generalized Fibonacci and Lucas sequences.
In this paper, we gave solutions of the Diophantine equations 16^{x}+p^{y}=z^{2}, 64^{x}+p^{y}=z^{2} where p is an odd prime, n is a positive integer and x,y,z are non-negative integers. Finally we gave a generalization of the Diophantine equation (4^{n})^{x}+p^{y}=z^{2}.
In this article, we consider the equation x^2+19^{m}=y^n, n>2, m>0. We find the solutions of the title equation for not only 2 \mid m but also 2\notdividesm.