Editing Quadric (section)

===Pythagorean triples===
This is well illustrated by [[Pythagorean triple]]s. A Pythagorean triple is a [[triple (mathematics)|triple]] <math>(a,b,c)</math> of positive integers such that <math>a^2+b^2=c^2.</math> A Pythagorean triple is ''primitive'' if <math>a, b, c</math> are setwise coprime, or, equivalently, if any of the three pairs <math>(a,b),</math> <math>(b,c)</math> and <math>(a,c)</math> is coprime.

By choosing <math>A=(-1, 0, 1),</math> the above method provides the parametrization
:<math>\begin{cases}
a=m^2-n^2\\b=2mn\\c=m^2+n^2 
\end{cases}</math> 
for the quadric of equation <math>a^2+b^2-c^2=0.</math> (The names of variables and parameters are being changed from the above ones to those that are common when considering Pythagorean triples).

If {{mvar|m}} and {{mvar|n}} are coprime integers such that <math>m>n>0,</math>  the resulting triple is a Pythagorean triple. If one of {{mvar|m}} and {{mvar|n}} is even and the other is odd, this resulting triple is primitive; otherwise, {{mvar|m}} and {{mvar|n}} are both odd, and one obtains a primitive triple by dividing by 2.

In summary, the primitive Pythagorean triples with <math>b</math> even are obtained as
:<math>a=m^2-n^2,\quad b=2mn,\quad c= m^2+n^2,</math> 
with {{mvar|m}} and {{mvar|n}} coprime integers such that one is even and <math>m>n>0</math> (this is [[Euclid's formula]]). The primitive Pythagorean triples with <math>b</math> odd are obtained as
:<math>a=\frac{m^2-n^2}{2},\quad b=mn, \quad c= \frac{m^2+n^2}2,</math>
with {{mvar|m}} and {{mvar|n}} coprime odd integers such that <math>m>n>0.</math>

As the exchange of {{mvar|a}} and {{mvar|b}} transforms a Pythagorean triple into another Pythagorean triple, only one of the two cases is sufficient for producing all primitive Pythagorean triples [[up to]] the order of {{mvar|a}} and {{mvar|b}}.