Is it possible to get a perfect square if you multiply three consecutive natural numbers?
Let the three consecutive natural numbers be denoted by n-1, n and n+1
By the conditions of the problem, we must have:
n-1> 0, so that:
Or, n> =2
Now, in accordance with the provisions governing the puzzle, there exists a positive integer p, such that:
n(n-1)(n+1) = p^2
Or, n(n^2 - 1) = p^2
This is possible iff:
n^2 - 1 = n*m^2, for some positive integer m
Or, n(n-m^2) = 1
Now, n is a natural number factor of 1, so that n =1. This violates the restriction n> =2, and accordingly it follows that it is not possible to get a perfect square if you multiply three consecutive natural numbers.