n to the n is greater than n factorial squared

3³ = 27 < 3!² = 6² = 36.

Bzzt! Wrong answer!

Not true. nⁿ ≤ n!² for all n a natural number.

Proof:

nⁿ = n * n * n * ... * n * n * n
n! = n * (n-1) * (n-2) * ... * 3 * 2 * 1
n! = 1 * 2 * 3 * ... * (n-2) * (n-1) * n
=> n!² = (n*1) * ((n-1)*2) * ((n-2)*3) * ... * (3*(n-2)) * (2*(n-1)) * (1*n)

This sequence contains n terms and each term is of the form ((n-i)*(i+1))
(n-i)*(i+1) = ni - i² + n - i
This is obviously greater than or equal to n for all i such that 1 ≤ i ≤ n
Hence, the sequence is the product of n terms, all greater than n. Hence it is greater than or equal to nⁿ.

QED

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Jurph: Fixed. Now it says ≤ all the way through, rather than <. Stupid typos.

...Except that i isn't restricted to the domain (1...i...n).

Where above, you said "the sequence is the product of n terms, all greater than n," you should have said, "the sequence is the product of n terms, all greater than or equal to n." The number of terms equal to n (2 of them) comes into play for extreme values of i.

You have to start your sequence with i=0 and end with i = (n-1) for your end-terms to be equal to (n*1) or (1*n), as follows: For i = (n-1),

(n - i) * (i + 1) = ni - i² + n - i

= 1 * n = n.

Likewise for i=0, the first term = n.

Since your first and last terms are equal to n, and you have n terms, you still need to prove that just one of those intervening terms is strictly greater than n to have solid proof.

If you only have 2 terms (and 2 is a natural number), the argument fails either way, and nⁿ = n!². That is, you can't prove one is greater or less for all cases, because of the equality case.

For n, any natural number, nⁿ and n!² have no provable relationship. Although your proof does refute the theory for all cases n > 2, you need this little caveat tacked on for n = 2. The trivial case n = 1 also disproves the theory above, but that is left as an exercise for the reader.

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Okay, the author above fixed a typo from "less than" to "less than or equal", so this mostly goes in /dev/null. Just a proof of the trivial case.