Skip to main content

[Community Question] Algebra-Precalculus: Find all integer solutions to $ x^3 - y^3 = 3(x^2 - y^2) $

One of our user asked:

The objective is to find all solutions to $$ x^3 - y^3 = 3(x^2 - y^2) $$ where $x,y \in \mathbb{Z}$.

So far I've got one pair of solution. Try $(x, y)=(0,0)$: $$ 0^3-0^3=3(0^2-0^2) \\ 0=0 \qquad \text{equation satisfied}$$

Another try $ (x, y) = (x, x)$ then $$ x^3 - x^3 = 3(x^2 -x^2) \\ 0 = 0 \qquad \text{equation satisfied}$$

But the above are just particular solutions. To find all $(x, y)$ pairs, I tried:

$$\begin{aligned} (x-y)(x^2+xy+y^2) &= 3(x+y)(x-y) \\ x^2+xy+y^2 &= 3(x+y) \end{aligned}\\ \begin{aligned} x^2+xy+y^2-3x-3y &= 0 \\ x^2+x(y-3)+y(y-3) &=0 \\ x^2 + (y-3)(x+y) &= 0 \end{aligned}$$

What now? Am I on the right track?


Labels:

Comments

Post a Comment

Popular posts from this blog

[Community Question] Calculus: Manifold with boundary - finding the boundary

One of our user asked: I have the manifold with boundary $M:= \lbrace (x_1,x_2,x_3) \in \mathbb R^3 : x_1\geq 0, x_1^2+x_2^2+x_3^2=1\rbrace \cup\lbrace (x_1,x_2,x_3) \in \mathbb R^3 : x_1= 0, x_1^2+x_2^2+x_3^2\leq1\rbrace$ and I need to find the boundary of this manifold. I think it is $\lbrace (x_1,x_2,x_3) \in \mathbb R^n : x_1= 0, x_2^2+x_3^2=1\rbrace$ , the other option is that the boundary is the empty set? I think the first is right? Am I wrong?
Post a Comment
Read more

[Community Question] Linear-algebra: Are linear transformations between infinite dimensional vector spaces always differentiable?

One of our user asked: In class we saw that every linear transformation is differentiable (since there's always a linear approximation for them) and we also saw that a differentiable function must be continuous, so it must be true that all linear operators are continuous, however, I just read that between infinite dimensional vector spaces this is not necessarily true. I would like to know where's the flaw in my reasoning (I suspect that linear transformations between infinite dimensional vector spaces are not always differentiable).
Post a Comment
Read more

Order of elements of the Prüfer groups $\mathbb{Z}(p^{\infty})$

Let $\mathbb{Z}(p^{\infty})$ be defined by $\mathbb{Z}(p^{\infty}) = \{ \overline{a/b} \in \mathbb{Q}/ \mathbb{Z} / a,b \in \mathbb{Z}, b=p^i$ $ with$ $ i \in \mathbb{N} \}$ , I wish show that any element in $\mathbb{Z}(p^{\infty})$ has order $p^n$ with $n \in \mathbb{N}$ . i try several ways but I have not been successful, some help ?? thank you
Post a Comment
Read more