Editing Diophantine set (section)

==Examples==
In the following examples, the natural numbers refer to the set of positive integers.

The equation

:<math>x = (y_1 + 1)(y_2 + 1)</math>

is an example of a Diophantine equation with a parameter ''x'' and unknowns ''y''<sub>1</sub> and ''y''<sub>2</sub>. The equation has a solution in ''y''<sub>1</sub> and ''y''<sub>2</sub> precisely when ''x'' can be expressed as a product of two integers greater than 1, in other words ''x'' is a [[composite number]]. Namely, this equation provides a '''Diophantine definition''' of the set

:{4, 6, 8, 9, 10, 12, 14, 15, 16, 18, ...}

consisting of the composite numbers.

Other examples of Diophantine definitions are as follows:
* The equation <math>x = y_1^2 + y_2^2</math> with parameter ''x'' and unknowns ''y''<sub>1</sub>, ''y''<sub>2</sub> only has solutions in <math>\mathbb{N}</math> when ''x'' is a sum of two [[square number|perfect squares]]. The Diophantine set of the equation is {2, 5, 8, 10, 13, 17, 18, 20, 25, 26, ...}.
* The equation <math>y_1^2 - xy_2^2 = 1</math> with parameter ''x'' and unknowns ''y''<sub>1</sub>, ''y''<sub>2</sub>. This is a [[Pell equation]], meaning it only has solutions in <math>\mathbb{N}</math> when ''x'' is not a perfect square. The Diophantine set is {2, 3, 5, 6, 7, 8, 10, 11, 12, 13, ...}. 
* The equation <math>x_1 + y = x_2</math> is a Diophantine equation with two parameters ''x''<sub>1</sub>, ''x''<sub>2</sub> and an unknown ''y'', which defines the set of pairs (''x''<sub>1</sub>, ''x''<sub>2</sub>) such that ''x''<sub>1</sub> < ''x''<sub>2</sub>.