Theorem 1: All integers solutions to $a^2+b^2=c^2$ are given by $a,b,c=(2 x y , x^2 - y^2 , x^2 + y^2)$
Proof: We use sagemath to parametrize the conic:
sage: K.<a,b,c>=QQ[]
sage: co=Conic(a^2+b^2-c^2);pa=co.parametrization();pa
(Scheme morphism:
From: Projective Space of dimension 1 over Rational Field
To: Projective Conic Curve over Rational Field defined by a^2 + b^2 - c^2
Defn: Defined on coordinates by sending (x : y) to
(2*x*y : x^2 - y^2 : x^2 + y^2),
Scheme morphism:
From: Projective Conic Curve over Rational Field defined by a^2 + b^2 - c^2
To: Projective Space of dimension 1 over Rational Field
Defn: Defined on coordinates by sending (a : b : c) to
(1/2*a : -1/2*b + 1/2*c))
Counterexample to Theorem 1:
$a,b,c=(9,12,15)$.
Proof: $15$ is not the sum of two integer squares.