Is it possible to get a perfect square if you multiply three consecutive natural numbers?
Consider (x - 1)x(x + 1) = x(x² - 1) = y².
Since the greatest common divisor of x and x² - 1 is 1, we have x = a², x² - 1 = b², for some natural numbers a and b.
But then (a²)² - b² = 1, which is impossible if a² and b are natural numbers.
Hence the product of three consecutive natural numbers cannot be a perfect square.
As a generalization, the product of any number of consecutive positive integers is never a perfect power. This was proved by Erdös and Selfridge in 1975.
Solution
