Is there any number other than 0 and 1, such that its factorial is a perfect square.?

Question

The factorial of a number x is defined as the product 1*2*3*...x.
The factorial of 0 is by convention 1. Is there any number other than 0 and 1 such that its factorial is a perfect square.

Answer 1

No, there is not.

If we take each integer that we use in a factorial, and break it down to its smallest primes, then in order for it to be a perfect square, we have to be able to divide all these prime into 2 identical groups.

This means that as we start searching, if n = prime (for n!), then we need n to appear in the group of smallest primes a second time. That won't happen until we reach 2n, but by that time we will have encountered a NEW prime. For example,
take 3. Now, 3 will not appear in the group of smallest primes again until we get to 6, but now 5 appears in the group, so we must get to at least 10, by which time we've encountered 7, etc.

It has already been shown that if P=prime, then the next prime (P2)<2*P.

QED