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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6 votes
2 answers
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Find the last two non-zero digits of $70!$

The question itself is quite straightforward, however, I am unable to get an exact answer to the problem. I have narrowed it down to four possibilities one from $\{18, 43, 68, 93\}$. The approach We ...
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0 votes
0 answers
33 views

Formula for $\binom{a}k - \binom{a-x}k$ [closed]

Is there a formula for $\displaystyle\binom{a}k - \binom{a-x}k$? One that perhaps reduces it down to a single binomial coefficient multiplied by something?
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1 vote
0 answers
35 views

Prove $ \lim_{z \to \exp \left(\frac{2 \pi i p}{m !}\right)} \sum_{n = 0}^{\infty} {z}^{n !} = \infty$ (Whittaker-Watson)

The following is a problem posed by Lerch in 1885. Problem: For $m \in \mathbb{Z}$ and $p = 0 , 1 , 2 , \ldots , \left(m ! - 1\right)$. Show that \begin{align} \lim_{z \to \exp \left(\frac{2 \pi i p}{...
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  • 571
-3 votes
0 answers
27 views

If $n,m$ are positive integers with gcd$(n,m)=1$, then $\frac{(m+n-1)!}{n!m!} \in \mathbb{Z}$ [duplicate]

If $n,m$ are positive integers with gcd$(n,m)=1$, then $$ \frac{(m+n-1)!}{n!m!} \in \mathbb{Z}$$
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0 votes
1 answer
62 views

Probability a factor is odd

Take the above question. In the solution it says there 18 factors of 2 in the number. But how do they work this out? I see no reasoning or intuition between the line of working 10 + 5 + 2 + 1 = 18. ...
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1 vote
1 answer
19 views

Bounding super exponential functions with factorial functions

I want to show that there exists polynomials $q(n)$, $p(n)$ with integer coefficeints such that $$ (q(n)!)^{p(n)} > 2^{2^n} $$ for all $n \in \mathbb{N}^+$. Intuitively this inequality seems to ...
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2 votes
1 answer
28 views

approximation for 'balls in bins' problem with upper restriction.

I am dealing with the famous problem of finding the how many integer non negative solutions there are for the equation: $ x_1+\cdots +x_l=n$ with the restrictions $ \forall i: 1\leq x_i \leq k.$ The ...
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  • 21
-1 votes
1 answer
75 views

What is $r-n+1$ equal to? [closed]

So I am working on factorials and can easily show that $$r = \frac{r!}{(r-1)!}$$ By expanding the factorial $$r! = r(r-1)\cdots(r-n+1)$$ Then dividing both sides by $(r-1) \cdots (r-n+1)=(r-1)!$ ...
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  • 27
1 vote
1 answer
56 views

How to calculate limit $\lim_{x\to\infty}\left(\lfloor x\rfloor+\frac{1}{2}\right)\ln x-\ln(\lfloor x\rfloor!)$

While uses plot on www.wolframalpha.com, i found a problem: The function $$f(x)=\left(\lfloor x\rfloor+\frac{1}{2}\right)\ln x-\ln(\lfloor x\rfloor!)$$ approximate the function $$g(x)=x-\frac{1}{2}\ln\...
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  • 1,339
-2 votes
0 answers
74 views

Is this the only twin prime of this kind?

Related to this question where $\ f(a,b)\ $ is defined. It is conjectured , but not proven , that $\ n=3\ $ is the only positive integer such that $\ n!-1\ $ and $\ n!+1\ $ are both prime. Is there a ...
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  • 79k
0 votes
0 answers
101 views

Find all triplets $(a,b,p)$ for $a^{p} = b! + p$ [duplicate]

This is the IMO 2022 number theory problem. Find all positive integer triplets $(a,b,p)$ such that: $$a^{p} = b! + p$$ where $p$ is prime. If the solution is uploaded by the official IMO website, I am ...
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  • 4,546
0 votes
1 answer
45 views

Show that the function is a bijection. [closed]

Let $A = \lbrace 1,2,6, 24, 120, .... \rbrace $. We define the following map $\alpha: \mathbb{N} \rightarrow A, \; \alpha(a) = a!.$ I'm not able to show by the definition that this map is bijective. I ...
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-2 votes
1 answer
50 views

Numerical Approximation - Finding roots [closed]

Is someone able to assist me or atleast point me in the right direction of finding the roots of the below formula: $$n^2−n+730\times\ln(0.5)=0$$ I have not done much numerical analysis/ approximation ...
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3 votes
1 answer
190 views

Solving $n!=x$ for $n$.

Trying to solve / approximate for $n$ in $$n!=x\tag 1$$ where $x$ is given, I started with Stirling's approximation $$n!\approx\sqrt{2\pi n}(n/e)^n\tag 2$$ Taking log and divide by $e$: $$\frac1{2e}\...
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2 votes
0 answers
60 views

Closed form for $\Gamma(n+\frac{2}{3})$

I'm looking for a good closed form for $\Gamma(n+\frac{2}{3})$ involving at most $\Gamma(\frac{1}{3})$. I know that $\Gamma(n+\frac{1}{3})=\Gamma(\frac{1}{3})\frac{(3n-2)!!!}{3^n}$ and my question is, ...
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2 votes
0 answers
72 views

Finite many primes for every positive integer $b$?

Consider the function $$f(a,b):=\sum_{j=0}^a (bj)!=1+b!+(2b)!+\cdots +(ab)!$$ Given a positive integer $b$ , are there always only a finite number of positive integers $a$ such that $f(a,b)$ is prime ...
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  • 79k
0 votes
0 answers
17 views

with respect to which partial order is this f monotonic?

My first task is to define a fixpointoperator f. It should represent the factorial-function: $fac = fix f$ $f = \lambda\ g \rightarrow \lambda\ x \rightarrow if\ x == 1\ then\ 1\ else\ x * g(x -1)$ ...
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  • 101
4 votes
2 answers
108 views

Finding $n$ such that $\frac{n+1}{n}$ < $\frac{\sqrt[n+1]{(n+1)!}}{\sqrt[n]{n!}}$

I have been thinking about this limit: $$\lim\limits_{n \rightarrow \infty}\frac{n}{\sqrt[n]{n!}} = e$$ Using a spreadsheet, I noticed that for $0 < n \le 150, \frac{n+1}{n} > \frac{\sqrt[n+1]{(...
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7 votes
1 answer
95 views

How would one go about summing the factorials?

I wish to sum the following series: $$ \sum_{r=1}^nr! $$ My initial thought was to first convert this in terms of the gamma function, and sum all of the integrals that this made like so: $$ \sum_{r=1}^...
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1 vote
0 answers
28 views

Reasoning about limits related to the prime number theorem

From the Prime Number Theorem, it follows that: $$\lim\limits_{n \rightarrow \infty}\sqrt[n]{n\#} = e$$ One of the standard definitions of $e$ as found here is that: $$e = \lim\limits_{n \rightarrow \...
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0 votes
4 answers
58 views

Evaluating $ \lim_{n \to \infty} {\sqrt[n]{\frac{(2n - 1)!}{n!}}} $

I have such limit to solve: $$ \lim_{n \to \infty} {\sqrt[n]{\frac{(2n - 1)!}{n!}}} $$ I understand that with the n-root I should go to $e^{\ln n}$, but the real problem cause factorials. What can I ...
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  • 109
1 vote
0 answers
42 views

How many simple substitution ciphers are there that leave no letter fixed

In the following question,           Suppose that you have an alphabet of 26 letters and a letter in the alphabet is said to be fixed if the encryption of the letter is the letter itself. How many ...
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4 votes
1 answer
58 views

Simplifying binomial sum of factorials

When evaluating a multi-indexed derivative (what exactly is not important now) I ran across the sum $$ \sum_{\nu \leq \mu} \frac{\mu!}{\nu!(\mu - \nu)!} |\nu|!|\mu - \nu|!=(|\mu| + 1)! $$ for $\mu$ a ...
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0 votes
0 answers
48 views

Indefinite integral of $\frac{(z+2 n-3)\text{!!}}{(z-2 n+1)\text{!!}}$

Any chance i can find a definite integral for this complex function? \begin{equation} f_n(z)=\frac{2^{-\frac{1}{2} ( \sin (\pi z) ) ( \sin (2 \pi n))+2 n-2} \pi ^{\frac{1}{2} ( \sin (\pi z) ) ( \...
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6 votes
2 answers
100 views

Calculate $\lim \sqrt[n] \frac{(2n)!}{(n !)^2}.$ [duplicate]

I want to calculate $$\lim_{n\to \infty} \sqrt[n] \frac{(2n)!}{(n !)^2}$$ According to Wolfram alpha https://www.wolframalpha.com/input?i=lim+%5B%282n%29%21%2F%7Bn%21%5E2%7D%5D%5E%7B1%2Fn%7D , this ...
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  • 2,377
0 votes
0 answers
93 views

How $5^{40} < 4^{60} < 27^{30}$, $8^{1/11} < 9^{1/10} < 10^{1/9}$, $e^{e} < \pi^{e} < e^{\pi}$, and $200^{100} < 200! < 100^{200}$?

The following four problems appeared in (UCLA) (University of California, Los Angeles) - GRE Preparation. $\mathbf{Problem} \space \mathbf{11}, \mathbf{Problem} \space \mathbf{12}, \mathbf{Problem} \...
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2 votes
1 answer
68 views

Express $\frac{1}{2n + 1}\,,\,n\in\mathbb{N}$ as a series of the form $\sum_{m = 0}^\infty\frac{a_m n!}{(n + m)!}\,.$

I'm interested in performing a sum, and as part of my summation, I need to express $$ \frac{1}{2n + 1}\,,\,n\in\mathbb{N}\,, $$ as a series of the form $$ \sum_{m = 0}^\infty\frac{a_m n!}{(n + m)!}\,. ...
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  • 235
1 vote
1 answer
88 views

Relation connecting $(3n)!$, $3^n$ and $n!$

Any idea on the relation for $(3n)!$ in terms of $3^n$ and $n!$ ? I have seen that there exist such relation for $(2n)!$ in terms of double factorials, ie. \begin{equation} (2n-1)!!= \frac{(2n)!}{2^n ...
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  • 173
0 votes
0 answers
18 views

Non-terminating representations of rational numbers in base factorial

What do non-terminating representations of rational numbers look like in base factorial? In particular, are there any representations $d$ of rational numbers such that $d$ is not terminating and the ...
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  • 7,458
0 votes
1 answer
33 views

How does $\frac{(2n)!}{(2(n+1))!}$ become $\frac{1}{2(n+1)(2n+1)}$?

Searching for the radius of convergence for: $$\sum\limits_{n=1}^{+\infty}\frac{3x^n}{(2n)!}$$ leads me to the limit: $$\lim\limits_{n\to+\infty}\frac{(2n)!}{(2(n+1))!}.$$ How to simplify: $$\frac{(2n)...
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1 vote
0 answers
30 views

proving an equation involving factorials is impossible

I have a calculation which ends up with the following form: $$a!(x!)^a=b!(y!)^b,$$ for $a,b,x,y$ non-negative integers. I would like to prove this is impossible in general unless $a=b$ and $x=y$ or ...
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  • 1,754
2 votes
1 answer
52 views

Creating binary information from a permutation

I created a random permutation with a deck of 52 unique cards. I want to transform this random permutation into a number that is represented in binary format with a length of $\lfloor \log_2(52!))\...
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17 votes
4 answers
593 views

Combinatorial proof of a simple inequality: $\left(\frac{n+1}{2}\right)^n \ge n!$

I want to prove the following inequality combinatorialy $$\left(\frac{n+1}{2}\right)^n \ge n! ,n \in \mathbb{N} $$ my attempts in this direction so far have been $$\left(\frac{n+1}{2}\right)^n \ge n! \...
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  • 279
1 vote
0 answers
90 views

How to prove that there are infinitely many positive integers $n$ such that both $n!+1,n!-1$ are composite numbers

According to Wilson's theorem if $p$ is a prime number we have $p\mid (p-1)!+1$ so if we prove that for infinitely many prime numbers $p$, $(p-1)!-1$ is composite we're done. Since $p-1=-1 \pmod p$ ...
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1 vote
0 answers
126 views

Is there integer $n>3$ such that both $n!-1$ and $n!+1$ are prime?

Is there integer $n>3$ such that both $n!-1$ and $n!+1$ are prime? For $n=3$ we know that $5=3!-1$ and $7=3!+1$ are prime.
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4 votes
4 answers
222 views

Is this a sound line of reasoning to conclude that $\sqrt[n]{n!} \sim \frac{n}{e}$?

In the paper Decomposable Searching Problems I. Static-to-Dynamic Transformations by Bentley and Saxe, the authors state without proof that $$\sqrt[n]{n!} \sim {\frac{n}{e}}\text.$$ I have a line of ...
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1 vote
1 answer
74 views

Sum of n + n(n-1) + n(n-1)(n-2) + ... + n!

This is to work out the time complexity of a computer science problem (write an algorithm to calculate the permutations of an array of n distinct integers). Various answers on leetcode say the sum ...
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0 votes
0 answers
38 views

When you sum the even terms and the odd terms of the binomial coefficient you get the same number, how can i prove this? [duplicate]

I am trying to that a function which outputs either 0 or 1 is balanced (has the same number of outputs which are 0 and which are 1). I've essentially proved that this is the case if the sum $\sum^{l}_{...
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5 votes
2 answers
113 views

Show that $ \binom{n}{i\;j\;k} \le \binom{n}{m\;m\;m} $

Is it true that for integers $i+j+k= 3m = n$ where $i , j, k , m , n\ge 0$ the inequality holds ? $$ \binom{n}{i\;j\;k} \le \binom{n}{m\;m\;m} $$ I tried to show $$ \frac{n!}{m!m!m!} \Big/ \frac{n!}{...
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  • 635
0 votes
0 answers
51 views

A Proof of Wilson’s Theorem (Without Using Modular Arithmetic)

Wilson’s Theorem says that any number $n$ is a prime number if, and only if, $(n−1)!+1$ is divisible by $n$. I came across this theorem and a proof for it in some obscure math textbook that I borrowed ...
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1 vote
1 answer
128 views

Find the least natural number such that its cube is less than its factorial.

Basically, find a natural number $k$ such that $k^3 > k!$ but $(k + 1)^3 < (k + 1)!$. Now I know that the answer is 5. The issue is in the deriving this through algebra. Here's what I did: $(k + ...
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  • 335
4 votes
2 answers
97 views

Sum of series in the form $\sum^{\infty}_{n=0}\frac{1}{(n+k)\cdot n!}$ for some k>0

I was recently given the series $$\sum^{\infty}_{n=0}\frac{1}{(n+3)\cdot n!}$$ and was told that it evaluated to a very nice number of $e-2$. However, I do not know how to arrive at that answer using ...
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  • 1,776
3 votes
3 answers
64 views

Proving by induction that $\frac{1}{n} \ge \frac{n!}{n^n}$

I have been trying to prove by induction that for all $n \ge 0, \frac{1}{n} \ge \frac{n!}{n^n} $ Here is my proof so far: Prove that for $ n = 1, \frac{1}{n} \ge \frac{n!}{n^n} $ $ \frac{1}{1} = 1 $ $...
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6 votes
3 answers
155 views

How to solve this limit with factorial? $\lim_{n\to \infty}\frac{n!}{n^n}(\sum_{k=0}^n\frac{n^k}{k!}-\sum_{k=n+1}^\infty \frac{n^k}{k!})$

I want to solve the limit $$\lim_{n\to \infty}\frac{n!}{n^n}\left(\sum_{k=0}^n\frac{n^k}{k!}-\sum_{k=n+1}^\infty \frac{n^k}{k!}\right)$$ This problem may be about Stirling's Approximation and Taylor ...
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  • 139
0 votes
0 answers
22 views

Are there any obvious flaws to representing relative factorials of each hyperoperation this way?

Note: I use the term Hyperfactorial (relative factorial) differently than used by wikipedia and wolfram. These sources use the term to describe what I am describing here as exactly (and only) the ...
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-3 votes
2 answers
44 views

How do I prove that, for any positive integer $n > 2$, $n^{n/2} < n!$ [closed]

I tried using Induction, but I couldn't prove the inequality. Any proof would work. Rewriting the question for clarity, here is its statement: For any positive integer $n > 2$, prove that $n^{n/2} &...
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1 vote
0 answers
55 views

A basic formula for the falling factorial

Suppose we have a family $\mathfrak{A}$ of some subsets of $\Omega$, which is locally finite, i.e. $$ X(\omega): = \sum_{A \in \mathfrak{A}} \mathbf{1}_{A}(\omega) <\infty $$ for all $\omega\in \...
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  • 23.7k
2 votes
1 answer
43 views

Calculating a sum in the coefficient of some generating function

I tried to calculate the coefficient of a generating function \begin{align*} \frac{1}{n} [u^{n-1}] e^{un} \frac{1}{(1-u)^2} \end{align*} and got to \begin{align*} \frac{1}{n} [u^{n-1}] e^{un} \frac{1}{...
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  • 390
2 votes
2 answers
87 views

Finding the maximum $k$ such that $(7!)!$ is divisible by $(7!)^{k!}\cdot(6!)!$

If $(7!)!$ is divisible by $(7!)^{k!}\cdot(6!)!$, then what is the maximum value of $k$? At first glance I couldnt think of anything except Legendre's formula for calculating powers of a prime in a ...
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2 votes
0 answers
44 views

Sum of factorials in 3D symmetrical random walk $\sum_{k=0}^n\binom{n}{k}^2\binom{2(n-k)}{n-k}$

I'm trying to prove that in a tridimensional symmetrical random walk all states are transient. Since it's an irreducible Markov chain, I only have to prove it for a generic state, since they are all ...
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