Pythagorean equation curve
In the Pythagorean equation 




x

2


+

y

2


=
1
,


{\displaystyle x^{2}+y^{2}=1,}


we would like to study its rational solutions, that is, its solutions




(
x
,
y
)


{\displaystyle (x,y)}

 such that x and y are both rational. This is the same as asking for all integer solutions
to 




a

2


+

b

2


=

c

2




{\displaystyle a^{2}+b^{2}=c^{2}}

; any solution to the latter equation gives us a solution 



x
=
a

/

c


{\displaystyle x=a/c}

, 



y
=
b

/

c


{\displaystyle y=b/c}

 to the former. It is also the
same as asking for all points with rational coordinates on the curve described by 




x

2


+

y

2


=
1


{\displaystyle x^{2}+y^{2}=1}

. (This curve happens to be a circle of radius 1 around the origin.)
