On the diophantine equation \(x^2-Ny^2=-1\) (Q1584144)

In the well-known monograph [\textit{L. J. Mordell}, ``Diophantine equations'', Academic Press, London (1969; Zbl 0188.34503)] the problem is formulated on establishing simple conditions for the solvability in integers \(x\) and \(y\) of the equation (1) \(x^2-Ny^2=-1\) where \(N\) is a natural number not equal to a perfect square. A necessary solvability condition for the equation (1) is formulated: Let the equation (1) be solvable in integers. Then there exists a representation of the number \(N\) as \(N=A^2+B^2\), where \(A\) and \(B\) are integers, \((A,B)=1\) and \(A\) is odd and quadratic residue modulo~\(N\).