# Thread: N! is never a perfect square

1. ## N! is never a perfect square

1st the problem: n! is never a perfect square for all n element of natural numbers.
The way I tried to attack it seems to work but it doesnt develop a proof. I noted that n! = n(n-1)...(2)(1) and sqrtn! is sqrt of all those individual terms and there will always be an irrational number in that set. Therefore it could never be a perfect square because an irrational number times anything but itself or its inverse gives an irrational number. My problem is i dont know how to prove there will always be an irrational number in that set. I looked at it broken out for up to like 15! and the number of irrational factors seems to be getting larger but that doesnt constitute proof.

2. Note that there is always a prime which divides $n!$ only once - namely the greatest prime less than $n$. Because suppose $p^2 \mid n!$ , where $p$ is the greatest prime less than $n$. Then $p<2p because $2p$ is the next multiple of $p$ after $p$ itself. But by Bertrand's postulate, there is a prime $q$ with $p, contradicting the maximality of $p$. $\blacksquare$

Your method would not work, because for instance $\sqrt 3 \sqrt 6 \sqrt 2 = 6$ is rational although none of the three factors are...

There may be a way without using the relatively heavy machinery which is Bertrand's postulate, but I doubt it.

3. ## Throwing in a curve

As a side comment, you can always express a positive odd integer factorial as a perfect square function.

E.g. you can have 3! as $1\cdot2\cdot3 = 1\cdot3\cdot2 = (2^2 - 1)\cdot2$

Second example: 5! = $1\cdot2\cdot3\cdot4\cdot5 = 1\cdot5\cdot2\cdot4\cdot3 = (3^2 - 4)(3^2 - 1)\cdot3$ which is enough to establish a pattern (you can use a sigma function to condense the expression).

My memory is vague, but I believe that you can do something similar with even integer factorials.

4. O.k. I didnt think my method was gonna work out. We actually havent proven Bertrands Postulate so i guess ill have to look it up and provide a proof of that inside the proof as justification. Thanks for your help you saved me a few sleepless nights before finals

5. Originally Posted by ChrisBickle
O.k. I didnt think my method was gonna work out. We actually havent proven Bertrands Postulate so i guess ill have to look it up and provide a proof of that inside the proof as justification. Thanks for your help you saved me a few sleepless nights before finals
I doubt you'll be able to prove it! It's a notoriously difficult theorem to prove by elementary means. Edros first became famous for his proof of the theorem.

6. Wow ya looking at that I dont think my professor would expect us to use that. He likes throwing out problems that have an elementary reason why they aren't true and watching us struggle with them. So I would think there would have to be another way to prove it. I just cant think of anything besides my original thought.

7. Well Bertrand's postulate is elementary. It's just a difficult, elementary theorem.

I'll see if I can think of another proof.

8. my teacher said its fine to use anything we can find. He said he just enjoyed watching us all struggle for a week.