Examples
The well known Pell equation
is an example of a Diophantine equation with a parameter. As has long been known, the equation has a solution in the unknowns precisely when the parameter is 0 or not a perfect square. In the present context, one says that this equation provides a Diophantine definition of the set
- {0,2,3,5,6,7,8,10,...}
consisting of 0 and the natural numbers that are not perfect squares. Other examples of Diophantine definitions are as follows:
- The equation only has solutions in when a is not a power of 2.
- The equation only has solutions in when a is greater than 1 and is not a prime number.
- The equation defines the set of pairs such that
Read more about this topic: Diophantine Set
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