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

The Question :

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If $n\geq2$, 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 $2n-2$. 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 $n\geq2$ 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 $n\geq N$, $n!$ 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 $S_n$ to somehow show that $|S_n|$ 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?


The Question Comments :
  • “Isn’t close at all”? All but finitely many cases is almost the entire problem: if you can get effective bounds on $N$ you only need to check finitely many cases. But the PNT is much stronger than Bertrand’s postulate.
  • I’ll take your suggestion and work on finding a bound tonight. However, the strategy of my prime number theorem proof was to prove what is essentially Chebyshev’s theorem just for primes. I’m not sure if this would count as not using Chebyshev’s theorem, since my (eventual?) proof could be extended to the all natural numbers if the $N$ for both of them is the same. I’ll check, and maybe they’ll be different, but hopefully there’s another way to prove this that uses more elementary methods.
  • *Another possible approach: Can you find an argument that, say, $\binom {2k}{k}$ is not a square?
  • could you please clarify what you mean by ‘doesn’t use’ Chebyshev’s theorem? Having tried this problem independently without using betrand’s postulate or chebyshev’s theorem, I’ve proved that the statement doesn’t always hold true if there isn’t a prime between (1/2)n and n and must hold true if (i.e. only if) chebyshev’s theorem is true. But would a proof that said that there must be a prime between (1/2)n and (3/4)n work? What I’m trying to say is, if a proof presented implies this theorem (which it has to), it could also be derived from it. Does that still make it valid as an alternative?
  • Seems to be very hard to do without Chebyshev. Notice that $6! = 5 \cdot(12)^2$, so we must know that the prime $5$ exists between $3$ and $6$.

The Answer 1

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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 $2$.

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