Pellian equations of special type

In this paper, we consider the solvability of the Pellian equation x 2 − ( d 2 + 1 ) y 2 = − m , $$\begin{array}{} \displaystyle x^2-(d^2+1)y^2 = -m, \end{array} $$ in cases d = nk , m = n 2l−1, where k, l are positive integers, n is a composite positive integer and d = pq, m = pq 2, p, q are primes. We use the obtained results to prove results on the extendibility of some D(−1)-pairs to quadruples in the ring Z [ − t ] $\begin{array}{} {\mathbb{Z}}[\sqrt{-t}] \end{array} $ , with t > 0.

PERIODS OF DUCCI SEQUENCES AND ODD SOLUTIONS TO A PELLIAN EQUATION

A Ducci sequence is a sequence of integer $n$-tuples generated by iterating the map $$\begin{eqnarray}D:(a_{1},a_{2},\ldots ,a_{n})\mapsto (|a_{1}-a_{2}|,|a_{2}-a_{3}|,\ldots ,|a_{n}-a_{1}|).\end{eqnarray}$$ Such a sequence is eventually periodic and we denote by $P(n)$ the maximal period of such sequences for given $n$. We prove a new upper bound in the case where $n$ is a power of a prime $p\equiv 5\hspace{0.6em}({\rm mod}\hspace{0.2em}8)$ for which $2$ is a primitive root and the Pellian equation $x^{2}-py^{2}=-4$ has no solutions in odd integers $x$ and $y$.

A diophantine problem in ℤ[1 + \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} \(\sqrt d\) \end{document})/2]

We characterize the existence of infinitely many Diophantine quadruples with the property D ( z ) in the ring ℤ[1 + \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\sqrt d$$ \end{document})/2], where d is a positive integer such that the Pellian equation x2 − dy2 = 4 is solvable, in terms of representability of z as a difference of two squares.