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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71
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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 ...
32
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0answers
972 views

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?...
17
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0answers
994 views

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 ...
13
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202 views

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, ...
13
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1answer
453 views

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 ...
11
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292 views

Inequality problem with factorials

I am not sure if this kind of "question" is welcome on MSE. Here is an olympiad-like problem that I would like to share with you: Let $a,b,c$ be nonnegative integers. Prove that $$ \frac{(a+b)!^2}...
10
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0answers
428 views

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 ...
10
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0answers
228 views

Primality of $n! +1$

I came across with a problem where I was required to examine primality of $n! +1$ (17! + 1 was the actual number). Although Wilson's Theorem could be manipulated for determining primality of $n! + 1$...
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376 views

Can $4+(2k)!$ ever be a perfect square over the integers?

Is there any pair of natural numbers $\{ k, m \}$ satisfying: $4+(2k)! = m^2$? I tried simplifying this into $$ 2^k k!(2k-1)!! = (m-2)(m+2) \text , $$ where !! denotes the double factorial, i.e., $1 \...
9
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137 views

Fraction of $1$s in binary representation of $n!$

I plotted a fraction of $1$s in binary representation of $n!$ (i.e. A079584/A072831) for $n$ from $1$ to $10^4$: It appears it might converge to some limit for $n\to\infty$. Can we (dis-)prove that ...
6
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74 views

Perfect powers nearest to factorials

Suppose, $n\ge 8$ is an integer. Let $s$ be the smallest non-negative integer such that $n!-s$ is a perfect power. Let $t$ be the smallest non-negative integer such that $n!+t$ is a perfect power. ...
6
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0answers
93 views

Can we conclude $n=p-1$?

Let $\ n\ $ be a positive integer and $\ p\ $ a prime number such that $$\ p^2\mid (2n)! + n! + 1$$ The only pairs $\ (n,p)\ $ I found so far are $(1,2)$ , $(2,3)$ , $(10,11)$ , $(106,107)$ , $(4930,...
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1k views

Swinging factorial and swinging constant

The Swinging factorial $n≀$ defined as $$n≀=\frac{n!}{\left\lfloor{n/2}\right\rfloor!^2}$$ is relatively common and I found some results on Google. But when $$\sum_{n=0}^{\infty}\frac{1}{n≀}$$is ...
6
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1answer
121 views

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 ...
6
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97 views

Possible formula for $ f(x) = \sum_{n=0}^{\infty}x^{-n!} $

I was wondering if we have a formula for the following function: $$ f(x) = \frac{1}{x^{0!}} + \frac{1}{x^{1!}} + \frac{1}{x^{2!}} + \frac{1}{x^{3!}} + ... = \sum_{n=0}^{\infty}x^{-n!} $$ (Like we ...
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131 views

Rolling a dice twice, (1,1) = 1/36 not 2/36? (Generalized Counting Principle confusion)

So I am confused over why the sample space of rolling a red die and green die results in (1,4) being different from (4,1), but there can only be one (1,1). Why can't there be (1 -red, 1-green) and (1-...
5
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233 views

Are these well known properties of binomial coefficients?

I apologize for the number of definitions. I did not know how to state these ideas any simpler. If anyone can help me simplify the definitions, I will be glad to shorten the details. Let: $x,n$ ...
5
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183 views

Prove that $\frac{m!}{200}\neq50x^2+51x+13$ and is my working correct?

Where $m$ and $x$ any real non-negative interger values. Prove that $$\frac{m!}{200}\neq50x^2+51x+13$$ Where $m!\geq20$ I understand that it may link into a trivial part of Brocard's problem. $$\frac{...
5
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0answers
172 views

Integer solutions for $\lceil{\sqrt{x!}}\rceil^2 - x! = y^2$

Consider the equation $\lceil{\sqrt{x!}}\rceil^2 - x! = y^2$. For $x \le 16$, the equation has the following integer solutions: $$ \begin{matrix} x = 0 & y = 0 \\ x = 1 & y = 0 \\ x = 4 &...
5
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1answer
73 views

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 ...
5
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1answer
244 views

Multiplying three factorials with three binomials in polynomial identity

I have checked the following identity (1) below for $n\leq 40$ with a computer. Let $(n)_k$ denote the falling factorial $n(n-1)\ldots (n-k+1)$, let $Z_n=\sum_{k=0}^n (n)_k x^{n-k}$, and finally let $...
4
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89 views

Prove that $2^{n}≤n!,\forall n\in\mathbb N$ using mathematical induction

I want to show that $2^{k-1}\leq k!\hspace{0.2cm} $ $\forall \hspace{0.2cm} k \in \mathbb{N}$. I tried this: For $n=1$, $2^{0}=1=1!$, the equality holds. Suppose that holds for n, then $2^{k}\leq(k+1)...
4
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1answer
127 views

Prove that $\varphi (v,\cdots, v)=\sum _{|\alpha |=k}\frac{k!}{\alpha !}v^\alpha \varphi^\alpha $

Firstly consider the following notations: Given any $\alpha =(\alpha _1,\cdots, \alpha _m)\in \mathbb{N}_0^m$ we define: $\color{red}{|\alpha|}:=\sum_{i=1}^m\alpha _i$ $\color{red}{\alpha !}:=\alpha ...
4
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0answers
85 views

"Non-trivial" solutions to equal products of consecutive integers

$\bullet\ \textbf{Question}$ One can find equivalent products of consecutive integers such as $$8\cdot9\cdot10\cdot11\cdot12\cdot13\cdot14=63\cdot64\cdot65\cdot66.$$ Other solutions of this have been ...
4
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78 views

Primes $p$ which satisfy $p \mid \sum_{i=1}^{p-1} i!$

This question is inspired from @Mathphile's problem: The value$\sum_{i=1}^n i!$ where $n \in \mathbb{N}$, is only semiprime for $n=3,4$ One can easily solve this conjecture by knowing that $9 \mid ...
4
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0answers
87 views

Meromorphic continuation of the multifactorial

The double factorial $z!!$, like the normal factorial function, can be extended to the complex plane using $$ z!! = 2^{z/2} \left(\frac{\pi}{2}\right)^{(\cos\pi z-1)/4} \left(\frac{z}{2}\right)! $$ ...
4
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0answers
84 views

Closed form from recurrence

We have $a(0)=a(1)=1$, $$a(n)=(-1)^{n-1}+2\sum\limits_{k=1}^{n-1}\binom{n}{k}(-1)^{n-k-1}a(k)$$ $$1,1,3,13,75,541,4683,\cdots$$ which has nice closed forms, ex. $$a(n)=\sum\limits_{k=0}^{n}k!{n\brace ...
4
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0answers
93 views

Closed form for the series $\sum^{\infty}_{x=2}\frac{1}{x!-1}$ and related ones

Does anyone know if there a closed-form expression for: A. $\displaystyle\sum^{\infty}_{x=2}\frac{1}{x!-1}$ B. $\displaystyle\sum^{\infty}_{x=0}\frac{1}{x!+1}$ C. $\displaystyle\sum^{\infty}_{x=0}...
4
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0answers
252 views

Prove that every positive rational number can be written in factorial base

Prove that for every $n \in \Bbb Q^{+}$, $\exists ! k \in \Bbb Z^{+}$ and a sequence of non-negative integers $\{a_m\}$ such that $a_1 \geq 0, a_k > 0$ and $0 \leq a_j < j, \forall j: \, 1 < ...
4
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0answers
99 views

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 ...
4
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0answers
165 views

Prove $n!≤(\frac{n+1}2)^n$ using AM-GM inequality.

The following is what I've done, but I feel it is too simple. There must be some lapse in logic I'm not seeing. Proof: By the AM-GM inequality, $$\sqrt[n]{n!}≤\frac{1+2+...+n}n=\frac{\frac{n(n+1)}2}...
4
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0answers
109 views

What is the fallacy in the proof (given below) that $(n-2)! \equiv 1$ when $n$ is a prime number?

By Wilson's theorem we know that if $n$ is a prime number then $(n-1)! \equiv n-1 \pmod n$ So, upon division by $n-1$ on both the sides we have $(n-2)! \equiv 1 \pmod n$ Edit 1: The teacher deducted ...
4
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0answers
118 views

Estimating $n!$ as $e \left(\frac ne \right)^n \le n! \le ne \left(\frac ne \right)^n$

I'm told that for $n \geq 2,$ $$\sum_{k=1}^{n-1} f(k) \leq \int_1^n f(x) \, dx \leq \sum_{k=2}^n f(k)$$ I am then asked to consider $\ln n! = \sum_{k=1}^n \ln k$ and show that for $n \geq 2$ $$n! \...
4
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0answers
99 views

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 ...
4
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0answers
121 views

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 ...
4
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0answers
112 views

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.....
4
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0answers
96 views

Which series of numbers effectively translates the factorial to the exponential function?

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 $$\...
4
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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 ...
4
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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 ...
3
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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 ...
3
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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 ...
3
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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: ...
3
votes
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}^\...
3
votes
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 ...
3
votes
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 ...
3
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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 ...
3
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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 ...
3
votes
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-\...
3
votes
0answers
48 views

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 $a^b$ ...
3
votes
1answer
171 views

Factorials, squares and Bertrand's postulate

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?

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