n!n! is never a perfect square if n≥2n\geq2. Is there a proof of this that doesn’t use Chebyshev’s theorem?

If n2, then n! is not a perfect square. The proof of this follows easily from Chebyshev’s theorem, which states that for any positive integer n there exists a prime strictly between n and 2n2. A proof can be found here.

Two weeks and four days ago, one of my classmates told me that it’s possible to prove that n! is never a perfect square for n2 without using Chebyshev’s theorem. I’ve been trying since that day to prove it in that manner, but the closest I’ve gotten is, through the use of the prime number theorem, showing that there exists a natural number N such that if nN, n! is not a perfect square. This isn’t very close at all. I’ve tried numerous strategies over the past weeks and am now trying to use the Sylow theorems on Sn to somehow show that |Sn| can’t be square (I haven’t made any progress).

Was my classmate messing with me, or is there really a way to prove this result without Chebyshev’s Theorem? If it is possible, can someone point me in the right direction for a proof?

Thanks!

Answer

Here is a way to do it. We’ll need De Polignac’s formula which is the statement that the largest k such that p^k divides n! is k=\sum_{i}\left\lfloor\frac{n}{p^i}\right\rfloor. Additionally, we’ll take advantage of the fact that the function \left\lfloor\frac{2n}{p}\right\rfloor-2\left\lfloor\frac{n}{p}\right\rfloor is only ever equal to 0 or 1.

Proof: Let’s start with even numbers. Suppose that (2n)!
is a square. Then \binom{2n}{n}=\frac{(2n)!}{n!n!}
is a square as well, and we may write \binom{2n}{n}=\prod_{p\leq2n}p^{v_{p}} where each v_p is even.
The critical observation is that for primes p>\sqrt{2n}
we have v_{p}=\left\lfloor\frac{2n}{p}\right\rfloor-2\left\lfloor\frac{n}{p}\right\rfloor, which must equal either 0
or 1, and since v_p is even, we conclude that v_{p}=0
for p>\sqrt{2n}. This will lead to a contradiction as \binom{2n}{n} cannot be composed of such a small number of primes – this would give impossibly strong upper bounds on the size of the central binomial coefficient.

For p\leq\sqrt{2n}, v_{p}=\sum_{i}\left\lfloor\frac{2n}{p^{i}}\right\rfloor-2\left\lfloor\frac{n}{p^{i}}\right\rfloor\leq\log_{p}2n
and so p^{v_p}=\exp(v_p\log p)\leq\exp(\log(2 n))= 2n, which gives the upper bound \binom{2n}{n}=\prod_{p\leq\sqrt{2n}}p^{v_{p}}\leq\left(2n\right)^{\sqrt{2n}}.
Expanding (1+1)^{2n}
there will be 2n+1
terms of which \binom{2n}{n}
is the largest. This implies that \binom{2n}{n}>\frac{2^{2n}}{2n+1}, and since \frac{2^{2n}}{2n+1}>(2n)^{\sqrt{2n}}=2^{\sqrt{2n}\log_2(2n)}
for all n> 18, we conclude that (2n)! is never a square.

To prove it for odd numbers, consider the quantity \frac{(2n+1)!}{n!n!}.
Observing that \left\lfloor\frac{2n+1}{p}\right\rfloor-2\left\lfloor\frac{n}{p}\right\rfloor
only takes the values 0 and 1 for odd p>1 we see that the above proof carries through identically with a slight modification at the prime 2.

Attribution
Source : Link , Question Author : Lincoln Blackham , Answer Author : AsukaMinato

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