We had an interesting discussion in our algebra department yesterday, about this innocuous-looking equation. I suggest you don’t try it, unless you’re equally happy with negative integer solutions or know something about elliptic curves!

As it turns out, the smallest positive integer solution has 80-digit numbers! Hence you won’t find these using brute-force computer searches without some insight on elliptic curves. If you want to read more about the mathematics behind this equation, read this delightful post on Quora by Alon Amit.