# Questions tagged [factorial]

Questions on the factorial function, $n!=n\cdot(n-1)\cdot...\cdot1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

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### Factorial of a matrix: what could be the use of it?

Recently on this site, the question was raised how we might define the factorial operation $\mathsf{A}!$ on a square matrix $\mathsf{A}$. The answer, perhaps unsurprisingly, involves the Gamma ...
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### What steps have been taken so far to solve Brocard's Problem?

The equation is $$n!+1=m^2$$ where $n$ and $m$ are natural numbers. Brocard's Problem asks whether there are solutions for n other than $4, 5, 7$. The only improvement I have found that people have ...
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### Is the occurence of two perfect numbers a coincidence?

The expression $$n^n+n!+1$$ is prime for the following non-negative integers $n\le 7\ 600$ : $$[0, 1, 2, 4, 28, 496]$$ if we assume $0^0=1$ The numbers $28$ and $496$ are perfect numbers. Is ...
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### Maxima and minima of $f(x)=\binom{16-x}{2x-1}+\binom{20-3x}{4x-5}$.

Find the maximum and minimum value of the function $f(x)=\binom{16-x}{2x-1}+\binom{20-3x}{4x-5}.$ ATTEMPT:- By A.M.-G.M. inequality, $\frac{a+b}{2}\ge\sqrt{ab}$, $\quad$ for $a,b\gt 0$ with equality ...
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### How can I use Stirlings inequality to prove this inequality?

Let $p,k$ be natural numbers with $p\ge k$, show that $$\frac{(p-k)!}{(p+k)!} \le \frac{1}{p^{2k}} \left(\frac{e}{2} \right)^{2k^2/p}.$$ The text where I come across this says to use Stirling's ...
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### Double Summation Over all subset of $\{1,2,…n\}$

In Benson's Book "Polynomial In variants of Finite Groups" It is claimed that(Without any proof): $$j! u_1u_2...u_j =\sum_{I \subseteq \{1,2,...,j\} } (-1)^I (\sum_{i \in I}u_i)^j$$ Where $I$ runs ...
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### Limit of $\frac{1}{n}\log({n\choose np})$ without using Stirling's formula

I am trying to evaluate the following limit: $$\forall p\in(0,1) ,\lim_{n\rightarrow \infty} \frac{1}{n}\log{n\choose \lfloor np \rfloor} =H(p),$$ where $\lfloor x\rfloor$ means the greatest integer ...
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### If $x$ and $y$ are positive integers then $\frac{(2x)!(2y)!}{x!y!(x+y)!}$ is an integer

If $x$ and $y$ are positive integers then $\frac{(2x)!(2y)!}{x!y!(x+y)!}$ is an integer I have to show that the proposition above is true for any $x,y\in\mathbb{Z^+}$ by means of Legendre's formula.....
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### Does the sum of reciprocals of primorials converge?

It is well known that the sum $$\sum _{{k=0}}^{\infty }{\frac {x^{k}}{k!}}$$ converges to $e^{x}$. In particular, for $x=1$ we have $\sum _{{k=0}}^{\infty }{\frac {1}{k!}}=e$. But what about the ...
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### Are there only finite many n-tupels of factorials summing up to a non-trivial power?

Let $n\ge 1$ be a positive integer and $S$ be the set of all $n$-tuples of positive integers $1\le a_1\le a_2\le \cdots \le a_n$ such that $$\sum_{j=1}^n a_j!$$ is a non-trivial power (a number $a^b$ ...