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.

328 questions with no upvoted or accepted answers
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36
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1answer
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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 ...
21
votes
1answer
541 views

An interesting formula for $\pi$

Looking through some old notebooks I found this monster of a formula: For any integer $r>1$, we have $$\pi=(-1)^{\left\lfloor\frac{r}{2}\right\rfloor-\left\lfloor\frac{2r-1}{4}\right\rfloor}\...
15
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0answers
911 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 ...
11
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1answer
353 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 ...
9
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0answers
203 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 $n! + 1$...
8
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0answers
120 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 ...
8
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0answers
253 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}...
7
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0answers
311 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 ...
6
votes
2answers
138 views

Prove that if $\Sigma_{i=1}^{n}(a_i)\le2^{n-4}$ then $a_2!a_3!\dots a_n!+1$ is not a power of $a_1$

The problem statement is Let $a_1,a_2,\dots,a_n$ be a strictly decreasing sequence of positive integers, with $a_1\equiv5\ (\text{mod }8)$. Prove that for all positive integers $n$, if $$a_1+...
6
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0answers
95 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 ...
5
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0answers
178 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
votes
1answer
102 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 ...
5
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0answers
91 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
votes
1answer
236 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
votes
1answer
49 views

Distribution of digits across all factorials

Show that the distribution of zeroes across all digits of all n! for $n\in \mathbb{N}$ converges to $ \frac{1}{6} $ and hence, $\frac{5}{54}$ for all other digits 1 through 9. In other words, let's ...
4
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0answers
62 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
69 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
49 views

Higher order factorials.

The function $y=x!$ can be drawn using the gamma functions, what could function would give higher order factorials like $y=x!!$ where $x!! = x*(x-2)*(x-4)*...*5*3*1$. I have this: $$\prod\limits_{n=0}^...
4
votes
1answer
114 views

Disprove $m!=100x^2+20x$ using an estimation for factorial.

$\newcommand{\floor}[1]{\lfloor #1 \rfloor}$ I have the equation $m!=100x^2+20x$ where $x$ and $m$ are real non-negative integers. I wish to disprove for when $m\geq20$ how can I do this? I had an ...
4
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0answers
84 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
96 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
99 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
99 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
88 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
59 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 ...
4
votes
1answer
81 views

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 ...
4
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0answers
110 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
votes
0answers
111 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
91 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
137 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
139 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
votes
0answers
67 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
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0answers
28 views

Meromorphic continuation of the multifactorial

The double factorial $z!!$, like the normal factorial function, can be extended to the complex plane using $$ z!! = 2^{z/2} \left(\frac{\pi}{2}\right)^{(\cos\pi z-1)/4} \left(\frac{z}{2}\right)! $$ ...
3
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0answers
82 views

Still unclear why Hanson's proof that $\text{lcm}(1,…,n)<3^n$ is not sufficient to resolve Legendre's Conjecture.

This is my second question on this topic. In my first question, mathlove quickly found the mistake. Correcting for the previous mistake, there is still a mistake in my analysis but I can't find it. ...
3
votes
0answers
94 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
2answers
145 views

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 ...
3
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0answers
46 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
0answers
79 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 < ...
3
votes
1answer
132 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?
3
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0answers
96 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}...
3
votes
1answer
115 views

Solving equations which includes factorials

I tried solve this question: If $x$ things can be arranged in $m$ ways, $x-2$ things can be arranged in $n$ ways and $x-6$ things can be arranged in $p$ ways and $m = 30np$, then find $x$. $m = 30np$...
3
votes
0answers
78 views

limit of a sequence involving double factorials

I have the following expression $$ x_N = N\left( \frac{(2N-1)!!}{(2N)!!} \right)^2$$ In order to prove that the sequence is convergent one can observe that $\frac{x_N}{x_{N-1}} = \frac{(2N-1)^2}{4N(N-...
3
votes
0answers
323 views

What is the path for solving $(20! \cdot 12!) \bmod 2012$?

I cannot solve this question: $(20! \cdot 12!) \bmod 2012$. I have tried finding the divisors but $2012$ equals $2 \cdot 2 \cdot 503$. $503$ is a prime number and I am stuck.
3
votes
0answers
38 views

Reasoning about a sequence of consecutive integers and factorials with hope of relating factorials to primorials

I am looking for someone to either point out a mistake or help me to improve the argument in terms of clarity, conciseness, and more standard mathematical argument. Let $x$ be an integer such that $x,...
3
votes
0answers
42 views

A discrete summation in probability theory

I have to solve this discrete sum for a statistics question. This sum for $$ \frac{1}{2^N} \sum_{d=-N,-(N-2),-(N-4)}^N \frac{\lvert d\rvert N!}{(\frac {N+d}{2})!(\frac {N-d}{2})!)}$$ To do this sum,...
3
votes
0answers
88 views

How to solve combination $C(n,k)$ for $k$?

I want to solve combination $C(n,k)$ for $k$ but don't know how. Suppose I have $n=10$ and $C(n,k)=15$. What will $k$ be?
3
votes
0answers
160 views

Solving equation involving factorials

I have this particular equation $\frac{(\alpha-1)!(\beta-1)!}{(\alpha+\beta-1)!} = \frac{\Gamma(p)(1+q)^{n+2p} 2^n}{q^{p}(2+q)^{n+p}}$. Now, given the values of $\alpha$ and $\beta$, I need to find ...
3
votes
1answer
55 views

Require assistance proving $n≥2 \Longrightarrow \frac{n!}{n^n} ≤ \frac{1}{2}^{\lfloor \frac{n}{2}\rfloor}$

Theorem: $n≥2 \Longrightarrow \frac{n!}{n^n} ≤ \frac{1}{2}^{\lfloor \frac{n}{2}\rfloor}$ Attempted Solution: We use induction. Additionally, we prove the stronger inequality omitting the floor ...
3
votes
0answers
346 views

Factorial of a large number and Stirling approximation

I'm trying to approximate the factorial of a large number with large precision. I know one can use the the Stirling approximation to do that with the formula: $$\sqrt{2\pi x} \left(\frac{x}{e}\right)^...
3
votes
0answers
258 views

The inverse of x!

what is the inverse of a factorial function? Its is not continuous but is modeled by the gamma function which is continuous so must have a inverse any research leads to the inverse gamma function that ...