prove that the product of five successive natural numbers can't be a perfect square.
Hi,
I had a thought on this problem... And started like you. But let's imagine n=pq.
Then, we can have p divides 2 (p=2) and q divides 4 (q=4). But yet n=8 and doesn't divide 1,2,3,4.
Whereas if we consider the consecutive numbers n-2,n-1,n,n+1,n+2, we have their product equal to :
And from this, it should be clear that n has to divide 4.
BUT there is still a problem. What happens if n is a square ? Or if n contains a square in its factors ???
We have to reconsider the equation
Consider the prime decomposition of
We have is divisible by n (if the product of the five consecutive numbers is a perfect square). So divides one of or , and the only possibilities this leaves for are and these can be eliminated by direct computation (I could add additional justification here but the reader should be able provide that for themselves - if I have got the argument right).
That should work, or have I still missed something?
CB