Questions on the factorial function, $n!=n\times(n-1)\times\cdots\times1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

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11
votes
2answers
685 views

when is $n!+10$ a perfect square?

When is $n!+10$ is a perfect square ? I have tried and found that only for $n=3$ is $n!+10$ a perfect square. Is there any other solution to this?
2
votes
0answers
241 views

Evaluating $\int e^{\Gamma(x)} dx $ and $\int \pi^{\Gamma(x)} dx $

I don't know how to solve these integrals: $$I_1 =\int e^{\Gamma(x)} dx $$ $$I_2 =\int \pi^{\Gamma(x)} dx $$ As a tenth grader I have no idea what the solutions could be. How would one go about ...
1
vote
1answer
146 views

Find all positive integers $a$, $b$, and $c$ for which $a \choose b$ $b \choose c$ = 2$a \choose c$

Find all positive integers $a$, $b$, and $c$ for which $a \choose b$ $b \choose c$ = 2$a \choose c$. Using the theorem ${n! \over k!(n-k)!} = {n \choose k}$ I simplified this down to $(a-c)! = 2(a-b)!...
1
vote
1answer
45 views

Divisibility problem using Wilson's theorem: $4(p-3)! + 2$ is divisible by $p$

Prove that $4(p-3)! + 2$ is divisible by $p$, where $p$ is an odd prime. Use Wilson's theorem. I am having trouble trying to bring it in the form where Wilson's theorem can be applied. Any help ...
0
votes
0answers
141 views

Equation with Sum of Factorial and Subfactorial

I am interested in finding solutions to the following equation $$x! + !x = a^3$$ where $x$ and $a$ are natural numbers and $!x$ is the subfactorial of $x$. I've found the solutions $x=1$ and $x=3$. ...
2
votes
0answers
121 views

How to differentiate $(x!)\uparrow\uparrow(!x)$?

I need help in differentiating the following expression with respect to x, which I recently came up with when trying to differentiate expressions involving subfactorials... $$(x!)\uparrow\uparrow(!x)$...
1
vote
2answers
141 views

Partitioning positive divisors of 100!

Is it possible to partition all positive divisors of 100! (including 1 and 100!) into 2 subsets so that each subset has the same number of integers and the product of all the divisors making up the ...
0
votes
0answers
135 views

Permutations Without Repetitions

Given the set [A,B,C,D] how many distinct ways can I order all four of the members of the set? I see distinct, as a unique set, therefore [A,B,C,D] and [D,C,B,A] ...
5
votes
6answers
188 views

The value of $ \int _{0}^{1}x^{99}(1-x)^{100}dx $ is

The value of $\int _{0}^{1}x^{99}(1-x)^{100}dx $ is Not able to do. I'm trying substituton. But clear failure. Please help.
1
vote
3answers
109 views

Use proof by induction to prove $ \frac{1}{n!}<\frac{1}{2^n-1} $ for all $n\geq 4$

Use proof by induction to prove that that $ \frac{1}{n!}<\frac{1}{2^n-1} $ for all $n\geq 4$, .\Base case: $$\frac{1}{4}=\frac{1}{24}\leq \frac{1}{2^4-1}$$ Inductive hypothesis: Assume there ...
0
votes
4answers
3k views

Gamma function proof of gamma $\;Γ(1/2) = \sqrt \pi\;$

So our teacher doesnt use the same demonstration as most other sites use for proving that gamma of a half is the square root of pi. I dont understand the demonstration from the first step because he ...
1
vote
1answer
93 views

Calculus over integers for derivative/integral of factorial?

One usually introduces the Gamma function to define a derivative of the factorial. However couldn't one define a derivative over integers like $$ f'(n) = \frac{f(n+1) - f(n)}{1} ? $$ Such a discrete ...
6
votes
1answer
245 views

Proving $11! + 1$ is prime

Prove that: $$11! + 1$$ is a prime number. Without computing the number (or factorial). Obviously, from Wilson's theorem, a number $n$ is prime if, $$(n-1)! + 1 \equiv 0 \pmod{n}$$ Since $n = ...
4
votes
2answers
97 views

Calculate $\sum_{n=1}^{\infty}\frac{(2n-1)!!}{(2n)!!\cdot 2^n}$

Calculate the sum $$\displaystyle \sum_{n=1}^{\infty}\frac{(2n-1)!!}{(2n)!!\cdot 2^n}$$ where $(2n-1)!!=1\cdot 3\cdots (2n-1)$, $(2n)!!=2\cdot 4 \cdots 2n$ Using Wolframalpha, the result is $\sqrt{...
0
votes
1answer
59 views

Is there any way to simplify this difference of factorials?

is there any way to simplify this expression or write it as a neat, concise formula? $$ \frac{(2m)!}{2m!} - \frac{(x+y)!}{x!y!} \cdot \frac{ [2m-(x+y)]!}{ (m-x)!(m-y)!} $$ Thank you!
3
votes
2answers
95 views

Stupid factorial question.

If i have $(3(n+1))!$ can I say: $(3(n+1))! = 3(n+1) \times (3n)!$ but if I expand by first multiplying the expression in the parenthesis; $(3(n+1))! = (3n+3) \times (3n+2) \times (3n+1) \times (...
2
votes
6answers
147 views

Prove by induction that $n^2<n!$

How can I show that $n^2<n!$ for all $n\geq 4$ Step 1 For $n=1$, the LHS=$4^2=16$ and RHS=$4!=24$. So LHS$<$ RHS. Step 2 Suppose the result be true for $n=k$ i.e., $k^2<k!$ Step 3 ...
3
votes
3answers
205 views

The ultimate formula to factor them all.

Context I am working on Integer factorization problem, I found a formula for factoring numbers, and I need your help to simplify it. First I will explain how I get there and then I present the ...
0
votes
2answers
62 views

A series specifying factorial [duplicate]

Is there any mathematical contraction of the following factorial series function? $$ F(n) = 1.1! + 2.2! + 3.3! + ...... + n.n! $$ I tried it by inspecting that $$ x.x! = x^2.(x-1)!$$ I need ...
1
vote
2answers
2k views

Prove that factorial grows faster than exponential function using limits [duplicate]

How can I prove that the factorial function ($n!$) grows faster than exponential functions (ex: $2^n$) using limits?
1
vote
0answers
66 views

Integral representation of simplified factorial division

I know that $\int_0^\infty{t^ne^{-t}}dt=n!$ from the gamma function, but I'm looking for a single integral which can represent $\frac{n!}{(n-m)!}=n(n-1)(n-2)\cdots(n-m+1)$ so that I don't need to ...
3
votes
1answer
99 views

Simple factorials

I've been doing some work with factorials and the normal way of calculating them is simply not working so well. When the numbers get really big, doing iterative multiplications is not viable and gets ...
7
votes
1answer
163 views

Prove $\limsup_{n\to\infty}|\{(p,q)\in T\times T,p!q!=n\}|=6$

Let $T$ be the set of nonnegative integers, I need to prove that $$\limsup_{n\to\infty}|\{(p,q)\in T\times T,p!q!=n\}|=6$$ It's really easy to show that $$\limsup_{n\to\infty}|\{(p,q)\in T\times T,...
11
votes
1answer
87 views

Does it follow that $(n!)^n$ divide $(n^2)!$

It is well known that $(n!)^2$ divides $(2n)!$. Does it follow that $(n!)^3$ divides $(3n)!$ and so on up to $(n!)^n$ dividing $(n^2)!$? If yes or no, could you provide the details behind the ...
1
vote
0answers
32 views

Rewriting Factorial Expressions [duplicate]

If I have an equation, say, $$ y = x! +1 $$ Is it possible to rewrite the equation in terms of $x$? For example, a simple algebraic equation might be $y = x^2$ --- and to rewrite it in terms of $x$, ...
5
votes
2answers
85 views

Prove that for any natural number $n>1$, we have $(2n)!<(n(n+1))^n$

Prove that for any natural number $n>1$, we have $(2n)!<(n(n+1))^n$ I tried using induction, but I failed in that approach, I rather found it was untrue, but on several case testing, I found ...
6
votes
3answers
150 views

Limit $\lim\limits_{n\to \infty} \sqrt [n]{\frac{(3n)!}{n!(2n+1)!}} $

First of all, sorry if something similar to this has been posted before (it's my first time in this web). I need to calculate the limit as $n\rightarrow \infty$ for this: $$\lim\limits_{n\to \infty} \...
1
vote
1answer
126 views

How to find the number of permutations with offset restriction

First question. Okay I have this problem that I've been trying to figure out for a while. I'm writing a computer program I need to quickly calculate the permutations of a set with 'n' elements with a ...
2
votes
2answers
160 views

Integer factorization: What is the meaning of $d^2 - kc = e^2$

I found an interesting behavior when placing the integer factorization problem in to geometry, I call it pyramid factoring. Lets assume we have $c$ boxes and we want to order them in to rectangle. ...
9
votes
5answers
180 views

Show $7!^{1/7} < 8!^{1/8}$

Show $7!^{1/7} < 8!^{1/8}$ So I know that the first step is to remove the radicals. So would I raise both sides to the power of 8 to get $({7!}^{1/7})^8 < 8!$. I am not sure where to go from ...
1
vote
2answers
73 views

Isn't $ n(n-1)(n-2)…(n-m+1) $ a factorial already?

Let $ m \ge 1 $ and $ n \ge 1 $ be integers Let $A$ be a set of size $m$ Let $B$ be a set of size $n$ How many one-to-one functions $f: A \rightarrow B$ are there? skipped stuff $$ n(n-1)(n-2)...(...
0
votes
1answer
225 views

How to solve Inequality with factorials

Im reading a book in Numerial analysis and I have the following which I dont understand involving inequalities and factorials, What i have is the following: $$\frac{1}{(2n+1)!(2n+1)} \leq 5*10^{-9}$$ ...
0
votes
1answer
123 views

Series increasing or decreasing with factorials

I have been working on some homework for calc 3 and my prof has put a couple sequences in which we must find if they are increasing or decreasing with factorials in them. I've googled and there are ...
0
votes
1answer
54 views

Computing $\mathrm{gcd} (100!, 3^{100})$

I am trying to compute $\mathrm{gcd}(100!,3^{100})$. I am still not sure how to reach an answer but I feel that Wilson's Theorem (i.e., $(p-1)!\equiv -1 \bmod p, p$ prime) and Fermat's Little theorem ...
0
votes
3answers
97 views

Cannot follow proof that $n! \leq en(n/e)^n$

prove that $n! \leq en(n/e)^n$ skip proof for base (n=1)... Assume it holds for $n-1$, verify for $n$. We have $n! = n* (n-1)! \leq n * e(n-1)(\frac{n-1}{e})^{n-1} $ by inductive assumption. we ...
3
votes
1answer
53 views

Prove for any integer $N$ that there exists $n > N$ where $n!-1$ is not a prime

I was thinking about Euclid's proof of the infinitude of primes and the fact that we could make the argument about $n!-1$ instead of $n!+1$ when I wondered if it would be easy to prove that for any $N$...
1
vote
2answers
48 views

Factorial formula problem [duplicate]

Prove that $(n-r)!(r!)$ divides $ n! $ i know its a factorial formula and it might be easy but i stuck .I tried induction to $n$ or analyzing the factorials but im missing something
0
votes
1answer
169 views

In how many ways can the word “WORD” be rearranged so that no letter is in its original position?

In how many ways can the word "WORD" be rearranged so that no letter is in its original position? The answer is $9$, but what is the formula for it?
1
vote
1answer
198 views

Finding the largest factorial with only three distinct decimal digits

I want to find the largest factorial whose decimal representation contains only three distinct digits. I am using the following Python code to compute the above, but no results up to 16000!: ...
1
vote
2answers
78 views

$x!=y^n$ for $x,y \neq 0,1$

A straightforward problem (find all integers such that $m!+3=n^2$) led me into thinking about the integers for which: $$x!=y^2$$ is true. I argued that other than the trivial case ($x!=1$) that this ...
1
vote
2answers
196 views

Inequality $(n!)^2\le \left[\frac{(n+1)(n+2)}{6}\right]^n$

Prove that $$ (n!)^2\le \left[\frac{(n+1)(n+2)}{6}\right]^n $$ holds for all $n\in\mathbb{Z^+}$. I tried induction but there's no obvious way to go from $n$ to $n+1$.
10
votes
3answers
480 views

Proving an identity involving factorials

I have stumbled upon the following statement and have verified it computationally for many $n$ (up to n=500, it took a long time for my computer to do out all of the math), yet I have no idea how to ...
0
votes
3answers
96 views

Simplifying factorials

I apologise for a repost, but my rep is not high enough to ask in a comment. But, in this question Simplify sum of factorials with mathematical induction I am confused how: $$(n+1)!-1+(n+1)(n+1)! = -...
3
votes
3answers
98 views

Proving $\frac{((2n)!)^2(i)!(j)!}{((n)!)^2(2i)!(2j)!}$ is an integer

I have verified this for many values of $n$, but I have no idea how to prove it. Does anyone know how I could go about showing that: $$\frac{((2n)!)^2(i)!(j)!}{((n)!)^2(2i)!(2j)!}$$ is an integer when ...
2
votes
1answer
35 views

Is there a way to express $(n-i)!(n-j)!(2i)!(2j)!$ in terms of $n$ and $r=i+j$?

I have been attempting to simplify the double sum: $$\sum_{i=0}^n \sum_{j=0}^n \frac{(-1)^{i+j} (2i+2j)!}{(n-i)!(n-j)!(2i)!(2j)!2^{i+j}(i+j)!}$$ And so what I am attempting to do is rewrite it in ...
2
votes
0answers
71 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
3answers
65 views

Proving $ n! \geq 2^{n-1} $

Prove that $$ n! \geq 2^{n-1}$$ for $n \geq 1$. My initial solution by induction goes like this. For $n = 1 : 1 \geq 1 $. Assuming that $$ n ! \geq 2^{n-1}.$$ Then for $n+1$, $$ (n+1)! = 2^{n+1-1}...
2
votes
3answers
65 views

Evaluate: $\sum_{k=2}^n\frac {n!}{(n-k)!(k-2)!}$

Evaluate: $$\sum_{k=2}^n\frac {n!}{(n-k)!(k-2)!}$$ Attempt $S_2=\frac {n!}{(n-2)!}$ $S_3=\frac {n!}{(n-3)!}$ $S_4=\frac {n!}{2(n-4)!}$ $\vdots$ $S_{n-1}=\frac {n!}{1!(n-3)!}$ $S_n=\frac {n!}{(...
3
votes
4answers
65 views

Showing $\binom{2n}{n} = (-4)^n \binom{-1/2}{n}$

Is there a proof for the following identity that only uses the definition of the (generalized) binomial coefficient and basic transformations? Let $n$ be a non-negative integer. $$\binom{2n}{n} = (-4)...
1
vote
1answer
53 views

Is there a closed form for this sequence?

I'm trying to find a closed form for the following sequence: $a$ $a(a-1)$ $a(a-1)(a-2)$ $a(a-1)(a-2)(a-3)$ The problem is, $a=\frac{1}{2}$. If it were some whole number, then I'd use $\frac{a!}...