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) = yn.
Since the greatest common divisor of x and x² - 1 is 1, we have x = an, x² - 1 = bn, for some natural numbers a and b.
But then (a²)n - bn = 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!
re(2): Solution
