Prove that there are infinitely many integer solutions to a diophantine equation

Question

Prove that there are infinitely many integer solutions to the diophantine equation: $(x-y)^7 = x^3y^3$

Answer 1

Let $x=y+h$. Then it's equal:

$$h^7=y^3(y+h)^3$$

Now let's try $y=ah$, you have:

$$h^7=a^3h^3(a+1)^3h^3$$

It's equal:

$$h=a^3(a+1)^3$$

Now you can write down infinitely many integer which satisfies equation:

$$y=a^4(a+1)^3$$

$$x=a^4(a+1)^3+a^3(a+1)^3=a^3(a+1)^4$$

For $a \in \mathbb{N}$.

Answer 2

Another solution: Let $x = \frac{k+1}{k} y$, then $$(x-y)^7=x^3 y^3 \Rightarrow \frac{y}{k^7}=\left(\frac{k+1}{k}\right)^3$$

$$\Rightarrow y=k^4(k+1)^3,\qquad x=k^3(k+1)^4$$