I am reading a Wikipedia article http://en.wikipedia.org/wiki/Diophantine_set. They say the diophantine equation
$x^2-d(y+1)^2=1$
has a solution in the unknows $x, y$ precisely when the parameter is $0$ or not a perfect square. $1$ is a perfect square so for $d=1$ the equation would not have a solution. But consider $x=1, y=-1$, these are the integer solutions of the equation.
What solution do they mean, some general one, not applicable just to one case? Or am I missing something simple?
The solutions to Diophantine equations are traditionally accepted from the whole domain of integers. However, in the context of specific theories like Peano Arithmetic or in some definitions like that of a Diophantine set, this domain of accepted solutions can be restricted to the natural numbers.