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.
The tangent line to $E$ at a point $(x_0,y_0)$ usually has equation $$(y-y_0)=m(x-x_0)\tag{1}$$ where $m$ is the value of $dy/dx$ at the point $(x_0,y_0)$. From $$y^2=x^3-n^2x\tag{2}$$ we get $$2y\frac{dy}{dx}=3x^2-n^2,$$ that is $$m=\frac{3x_0^2-n^2}{2y_0}.$$ To find the intersection points of $E$ with the tangent, putting $(1)$ into $(2)$ gives $$x^3-m^2x^2+(\cdots)x+(\cdots)=0.\tag{3}$$ Let's not worry what the coefficients of $x^1$ and $x^0$ are. The sum of the three roots of $(3)$ are $m^2$. But these three roots are $x_0$, $x_0$ and $x_1$. The root $x_0$ is repeated as the line is a tangent to the curve there. So $$x_1=m^2-2x_0=\frac{(3x_0^2-n^2)^2}{4y_0^2}-2x_0 =\frac{(3x_0^2-n^2)^2}{4(x_0^3-n^2x_0)}-2x_0$$ etc. Now you have to massage $x_1$ and $x_1\pm n$ into a form that makes it clear that all of them are square...