# 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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### Evaluating $\sum\limits_{x=2}^\infty \frac{1}{!x}$ in exact form.

$$\large \text{Introduction:}$$ We know that: $$\sum_{x=0}^\infty \frac{1}{x!}=e$$ But what if we replaced $x!$ with $!x$ also called the subfactorial function also called the $x$th derangement number?...
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### Understanding Ramanujan's approach in his proof of Bertrand's Postulate

I've been reading through Ramanujan's proof of Betrand's Postulate and I'm not clear why he didn't state his proof in terms of $\varphi(2x) - \varphi(x)$ What would be wrong with this approach for ...
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### Is 7 the only solution for $n!+1 \equiv 0 \mod 10n+1, n \in \mathbb{Z}^{+}$?

I've tested each $0<n<10^5$, it seems that 7 is the only solution for $n!+1 \equiv 0 \mod 10n+1$. But I can't prove it correct or wrong. I know that no solution exists if $10n+1$ is composite, ...
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### Double factorial as a sum

I believe the following equality to hold for all integer $l\geq 1$ $$(2l+1)!2^l\sum_{k=0}^l\frac{(-1)^k(l-k)!}{k!(2l-2k+1)!4^k}=(-1)^l(2l-1)!!$$ (it's correct for at least $l=1,2,3,4$), but cannot ...
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### Smallest prime of the form $68^k+k!+1$?

Let $f(n)$ be the smallest integer $k\ge 1$ such that $$n^k+k!+1$$ is prime or undefined if no such $k$ exists. I determined the values $f(n)$ for the even numbers $2,4,6,\cdots$ and $f(56)$ turned ...
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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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### 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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We have the relation of the Bernoulli numbers $$B_{2n} = (-1)^{n+1}\frac {2(2n)!} {(2\pi)^{2n}} \left(1+\frac{1}{2^{2n}}+\frac{1}{3^{2n}}+\cdots\;\right).$$ For $n>1$, the right hand sum $$\... 0answers 141 views ### Trying to show that \ln(x!) - \ln(\lfloor\frac{x}{2}\rfloor!) - \ln(\lfloor\frac{x}{3}\rfloor!) - \ln(\lfloor\frac{x}{6}\rfloor!) \ge \psi(x) I've been told that the approach below will not work. I would be interested if someone could help me to understand what will go wrong. Let:$$\psi(x) = \sum\limits_{p^k \le x} \ln p$$So that (see ... 0answers 147 views ### Generalizing Jitsuro Nagura's result: my resulting upper bound for the second chebyshev function is too low. What am I doing wrong? I've been reading through Jitsuro Nagura's classic proof that there is a prime between x and \frac{6x}{5} and it seems to me that it should be possible to improve on his upper bound for the second ... 0answers 53 views ### Showing that the partial sums of this alternating series with factorials do not vanish before limit I am trying to prove (or disprove) the non-vanishing of the finite summation$$\sum_{k=0}^n (-1)^k \frac{2k+1}{\sqrt{(n - k)!(n+1+k)!}} \neq 0 \quad \forall n < \infty.$$Alternatively, that the ... 0answers 120 views ### Diophantine with factorials This is a problem I encountered on a competition Discord server, apparently, there is an elementary solution, but I'd honestly be fine with any solution. Wolfram Alpha solves the problem, here's the ... 0answers 54 views ### Counting degree n polynomials that are divisible by n! Let C_n denote the number of (monic) degree n polynomials P(x) = \prod_{t=1}^{n}(x-t) \pmod {n!} such that n! | P(x) for all integers x. Here are the first four examples: ... 2answers 65 views ### Differential Equation Solution By Power Series Solve (1 + x)y' = py;\ \ \ y(0) = 1, where p is an arbitrary constant. First I plugged in the guess y = \sum_{n = 0}^\infty a_n x^n: (1 + x)(\sum_{n = 0}^\infty a_n x^n)' = p\sum_{n = 0}^\... 0answers 82 views ### Factorization into factorial and combination? Is there a way or a general formula to write down all factorizations of n!\ ( ∀ n ∈ N) into the factors which themselves are in factorial (m!) and combination ( C(r,k) ) forms? And also is ... 0answers 90 views ### Evaluate \lim_{n \to \infty} \frac{n \# \cdot n!}{p_{n}\#} and estimate the area under f(x)=\frac{x \# \cdot x!}{p_{x}\#} Consider the follow expression:$$\displaystyle{\frac{n \# \cdot n!}{p_{n}\#}}, n \in \mathbb{Z^*}$$In this thread, the notation will mean the following: n\# is the primorial of n, defined as ... 0answers 99 views ### a version of the famous n!+1=m^2 problem It is still unknown if n!+1=m^2 has only 3 solutions, i.e. for n=4,5,7. n!+1 being a perfect square https://en.wikipedia.org/wiki/Brocard%27s_problem What about my problem: Are there ... 0answers 91 views ### Prove that f(n)=n^{2007}-n! is an Injective Map If f: \mathbb{N} \to \mathbb{Z} defined as f(n)=n^{2007}-n! Then Prove that it is an Injective function My try: According to the definition of Injective function: If p,q \in \mathbb{N} and ... 0answers 102 views ### Find region of convergence of double power series How can i calculate the region of convergence of this double power series ?$$ S(x,y)=\displaystyle{ \sum_{n=0}^{\infty} \sum_{k=0}^{\infty} \frac{(n-\frac12)!\>(k-\frac12)!\>(\frac{n}{2}+k-\...
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 $a^b$ ...
With Bertrand's postulate at hand, it is easy to see that $n!$ is never a square for $n\ge 2$ (because there is a prime between $n/2$ and $n$). But are there more elementary proofs of that fact?