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:
n> 1
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.

Posted by K Sengupta on 2007-06-29 12:19:42