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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?

Dávid Natingga
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    They mean a solution in the natural numbers. – user7530 Jul 15 '13 at 03:58
  • Is the solution required to be in natural numbers for all the types of Diophantine equation? Or this is something not defined and we need to specify for every Diophantine equation separately where the solutions are expected to be found? – Dávid Natingga Jul 15 '13 at 04:05
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    In the definition of Diophantine set, it is built in that the variables are non-negative. That is not a common restriction elsewhere, it is for historical reasons in logic. The formal theories (like Peano Arithmetic) tend to be theories of the non-negative integers. That's also why the rather weird $y+1$ instead of $y$, – André Nicolas Jul 15 '13 at 05:22
  • The comments answer my question completely, either of you can summarize/repost them as an answer. – Dávid Natingga Jul 15 '13 at 14:08

1 Answers1

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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.

Dávid Natingga
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