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is $\sqrt{n!}\notin\mathbb{Z}$ for $n>1$ true? [duplicate]

Is it true that $\sqrt{n!}\notin\mathbb{Z}$ for $n>1$? This is what I did: By induction: for $n=2$ its trivial ($\sqrt{2}$ is irrational). Suppose its true for some $n\in\mathbb{N}$, then the ...
29 views

Is there any number which is a perfect square of some number as well as factorial of some number. [duplicate]

Is there any number which is a perfect square of some number as well as factorial of some number? That is, let x be such a number, then x can be expressed as ...
4k views

Is $\sqrt{n!}$ a natural number?

I'm new here (on Mathematics Stack Exchange). Also, I'm a 10th grade student not a math expert. So, my question is whether, $$\sqrt {n!}$$ comes in the set of the Natural Numbers. There were ...
5k views

731 views

Factorials and their perfect squares

How many positive factorials are also perfect Squares. So for example $1!=1=1^2$. How many others exist other than 1? Is there any way to prove this?
443 views

Prove: $\lfloor \sqrt{n!}\rfloor\nmid n!$ [closed]

$n$ is an integer greater than $7$. How does one go about proving that $\lfloor \sqrt{n!}\rfloor\nmid n!$.
Prove that $L_1$ = $\{1^m :$ m is not a perfect square$\}$ is not regular using Pumping Lemma
I'm trying to prove that the Language $L_1$ = $\{1^m :$ m is not a perfect square$\}$ is not regular. I proved before that L = $\{1^m :$ m is a perfect square$\}$ is not regular, I thought that I ...
Occurrence (parity) of primes in $n!$
It is known that $n!$ can not be a perfect square for $n\geq 1$. This means that in the prime decomposition of $n!$, one of the prime occurs odd number of times. This leads to following two questions: ...