The question is: Find a whole number x such that x+3 and x^2+3 are both perfect cubes, or show that this is impossible.
Since it was a question for a free scoop at an ice cream shop, I assumed it wouldn't require a lot of number theory, and could maybe be answered via a short and clever "trick".
But initially, I had trouble finding the trick. My first answer was a kind of clunky answer:
Later, I found what was probably the intended "trick". Here's a hint:
Finally, and here's where things might get a little more mathematically interesting:
Equivalently, I believe x^2 = y^3 - 3 has no integer solutions. This is an instance of Mordell's equation (except x and y are labeled in the opposite way to the usual convention) with n = -3, and some Googling seems to reveal that this is one of the cases of Mordell's equation that has no solutions.
I wonder if there's a reasonably self-contained proof that x^2 = y^3 - 3 has no integer solutions. It seems that there are for some instances of Mordell's equation, but others may be more subtle.