<s>[INST] Find three consecutive positive integers a,b and c such that a+b is a perfect square and b+c is a perfect cube. Find three consecutive positive integers a,b and c such that a+b is a perfect square and b+c is a perfect cube.
Any ideas on how to begin?
 [/INST] For all $x,y \in \mathbb{N}$, $x^3-y^2=2$ according to the given question, where $x=a+b$ and $y=b+c$. However, $x^3 \equiv 0,\pm1 \pmod4$ and $y^2 \equiv 0,1 \pmod4$.
So to satisfy the given equation, the following must hold: $$x^3-y^2\equiv 2 \pmod4$$
For this means that the required $x,y$ are such that $x^3 \equiv -1\equiv 3 \pmod4$ and $y^2 \equiv 1 \pmod4$. That is, the required forms are $$x^3=4n+3$$ $$y^2=4n+1$$
See if this helps.
</s>