I am self studying Elliptic curves / Algebra from book "Elliptic Curve Number theory And Cryptography second edition" I got stuck trying to do these 2 proofs questions.
Let $n$ be an integer. Show that if $x, y$ are rational numbers satisfying $y^ 2 = x^3 − n^2x$, and $x \neq 0, ±n$, then the tangent line to this curve at $(x, y)$ intersects the curve in a point $(x_1 , y_1 )$ such that $x_1 , x_1 − n, x_1 + n$ are squares of rational numbers.
Actually this question got a first part that was easy to answer:
Show that if $x, y$ are rational numbers satisfying $y^2 = x^3 − 25x$ and $x$ is a square of a rational number, then this does not imply that $x + 5$ and $x − 5$ are squares.
It was easy to proof by 1 counter example $x = 25/4$ which was already given as a hint.
The second question
Let $(x, y)$ be a point on the elliptic curve E given by $y^2 = x^3 + Ax + B$. Show that if $y = 0$ then $3x^2 + A \neq 0$. (Hint: What is the condition for a polynomial to have x as a multiple root?)
Besides the solution I would appreciate any tips on how to think about doing proofs in general in ECC or Abstract algebra.