Is it possible to get a perfect square if you multiply three consecutive natural numbers?

(In reply to

re: Solution by Richard)

Yes, that is more direct.

However, consider (x - 1)x(x + 1) = x(x² - 1) = y^{n}.

Since the greatest common divisor of x and x² - 1 is 1, we have x = a^{n}, x² - 1 = b^{n}, for some natural numbers a and b.

But then (a²)^{n} - b^{n} = 1, which is impossible if a² and b are natural numbers.

And, using the same approach, we have proved the more general result that no product of three consecutive natural numbers can be a perfect power!