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

learn more… | top users | synonyms

3
votes
1answer
65 views
+200

Find a convergent solution for $a$

Find a value for $a$ in which the following sum converges. $$a+a!+(a!)!+((a!)!)!+\cdots$$ I know that there are no solutions if you only look at $a\in \Bbb{R}$, but are there any solutions if you ...
11
votes
3answers
920 views

Intermediate digits of 34!

Problem: Given that $34!=295232799cd96041408476186096435ab000000$. Find $a, b, c, d$. $a, b, c, d$ are single digits. I am able to find $a$ and $b$ but cant find $c, d$. I did the prime factorisation ...
0
votes
1answer
30 views

Factorial and Multiplication

KISS: Is there anything I could do with $$ {xN\choose yN}$$ Given any size $N$, I would like to see how many ways there are to choose a fraction $yN$ out of $xN$. In factorials, that is ...
5
votes
4answers
90 views

Proving $n! = n \Rightarrow (n = 1 \quad or \quad n = 2)$

I want to know whether my proof is correct. Any elegant proofs are welcome. $n\in\mathbb{N}.\quad$Prove $ (n!=n) \Rightarrow (n=1\quad or\quad n=2)$ $ (n!=n) \Rightarrow (n=1\quad or\quad n=2) ...
-1
votes
1answer
38 views

Find the natural number $n>2$ such that $\frac{n!}{(n-1)!} + \frac{n!}{3!(n-3)!} = 2\frac{n!}{2!(n-2)!}$ [on hold]

I'm unsure how I'm supposed to solve the equation: $$\frac{n!}{(n-1)!} + \frac{n!}{3!(n-3)!} = 2\frac{n!}{2!(n-2)!} $$ given that $n>2.$
0
votes
1answer
29 views

Can Stirling's approximation be used to obtain lower and upper bound for $\pi(x)$?

The Willan's formula is given as follows (taken from Ribenboim's Little book of bigger primes): $$ \pi(x)=\sum_{j=2}^{x}f(j) \text{ where } ...
1
vote
3answers
82 views

How do you simplify $n!-(n-1)!$ [on hold]

I'm unsure how to simplify the expression $n!-(n-1)!$. Working as well as the final answer would be preferable.
1
vote
1answer
78 views

Solve the factorial equation $x! = c$

How to find the value of $x$ which its factorial for example equals to 100 ? $x! = 100 $ $x= ?$
58
votes
1answer
3k views

$n!$ is never a perfect square if $n\geq2$. Is there a proof of this that doesn't use Chebyshev's theorem?

If $n\geq2$, then $n!$ is not a perfect square. The proof of this follows easily from Chebyshev's theorem, which states that for any positive integer $n$ there exists a prime strictly between $n$ and ...
2
votes
1answer
22 views

Reason of dividing to n! ( repetition ) on Permutations with Repetitions

I'm trying to figure out the reason of diving the number of permutations by the number of repetitions (in factorial). Shouldn't it be without the factorial? I don't get why are there is a factorial in ...
-1
votes
2answers
59 views

Prove by induction and recursion that $n!=n(n-1)(n-2)…(3)(2)(1). $

Prove by induction and recursion that $n!=n(n-1)(n-2)...(3)(2)(1). $ We can start with the definition of factorial with recursion: $$n!= \left\{\begin{align}1\quad \text{for}\quad ...
0
votes
1answer
59 views

Permutations excluding repeated characters

I'm working on a Free Code Camp problem - http://www.freecodecamp.com/challenges/bonfire-no-repeats-please The problem description is as follows - Return the number of total permutations of the ...
18
votes
7answers
4k views

Stirling's formula: proof?

Suppose we want to show that $$ n! \sim \sqrt{2 \pi} n^{n+(1/2)}e^{-n}$$ Instead we could show that $$\lim_{n \to \infty} \frac{n!}{n^{n+(1/2)}e^{-n}} = C$$ where $C$ is a constant. Maybe $C = ...
40
votes
15answers
4k views

Which is larger? $20!$ or $2^{40}$?

Someone asked me this question, and it bothers the hell out of me that I can't prove either way. I've sort of come to the conclusion that 20! must be larger, because it has 36 prime factors, some of ...
7
votes
1answer
127 views

Partial sums of falling factorials

I want to know if there exists some way, approximate or exact, to do a partial sum of falling factorials of the kind: $$\sum_{k=i}^{n}(a+k)_{h}$$ where all are constants. And I'm interested too in ...
1
vote
1answer
52 views

Why is $\frac{1}{x} \sum_{n=1}^x \ln (n) \sim \ln(x) - \gamma$

I was playing with some functions and decided I wanted to see at which point the factorial of $x$ became bigger than $e^x$. I set them equal to each other and after doing some algebra I ended up with ...
0
votes
1answer
39 views

Finding a positive lower bound of hte sequence $\frac{\sqrt[n]{n!}}n$

I am given a sequence {(n'th root of n!)/n}. Can I show that the sequence is bounded below by a real no. which is greater than 0, by not calculating the limit of it....???thank you
2
votes
1answer
44 views

Infinitely many correct solutions to equation? [duplicate]

I conjecture that there are infinitely many correct solutions to this equation: (Where we are assuming $a,b \in \Bbb{N}$) $$a!+1=b^2$$ I chose to list the first three solutions below: $4!+1=5^2$ ...
2
votes
1answer
23 views

How to find the next higher combination out of a fixed group of digits?

I have a group of contiguous digits ordered from smallest to highest: 1234. I want a formula (in case it exists) to find the next closer higher combination of the same digits. In this example the next ...
0
votes
1answer
50 views

Solve limit using Stolz's theorem: $\lim_{n\to\infty} \frac{n}{\sqrt[n]{n!}}$ [closed]

Solve this limit usinig Stolz's theorem. Any help?! $$\lim\limits_{n\rightarrow \infty} \frac{n}{\sqrt[n]{n!}}$$
1
vote
0answers
47 views

Why do the mathematicians stated $0!$ to be $1$? [duplicate]

My question is very simple, if just as we say $5! = 120, 4! = 24,$ how can we say that $0! = 1$? Why did the ancient mathematicians conventionally consider $0!$ to be $1$? Then there's coming lot of ...
11
votes
1answer
149 views

Proof that the factorial is nonelementary

Is there a proof that the factorial function $!:\mathbb N\to\mathbb N$ is nonelementary? If it were equal to an elementary function (call it $P(n)$), then it would extend the factorial function to ...
3
votes
4answers
98 views

Why does $\sqrt{3x} \left(\dfrac {x}{2} \right)^x$ approximate $x!$ pretty well?

I was just messing around and trying out things in the desmos calculator and found that $\sqrt{3x} \left(\dfrac {x}{2} \right)^x$ is pretty close to $x!$ most of the time, here is a graph. Why does ...
45
votes
6answers
4k views

Finding the limit of $\frac {n}{\sqrt[n]{n!}}$

I'm trying to find $$\lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}} .$$ I tried couple of methods: Stolz, Squeeze, D'Alambert Thanks! Edit: I can't use Stirling.
2
votes
1answer
66 views

How is this equation evaluated $\binom {n}2 = \frac{n^2}{2}$?

I would like to know how $\binom {n}2 = \dfrac{n^2}{2}$ works out while I'm reading a proof on this page. I have tried several ways, but I couldn't. i.e. we knew that combinatorics formula that ...
1
vote
1answer
68 views

Wilson's Theorem Factorial

I need to prove that $ (1 \cdot 3 \cdot 5 \dotsm 2009)^2 - 1 \equiv 0 \pmod{2011}$ By modular simplification, I need to prove that $(3 \cdot 5 \cdot 7 \dotsm 2009) \equiv 1 \pmod{2011}$ I know that ...
0
votes
4answers
636 views

The sum of the factors of 9! [closed]

The sum of the factors of 9! which are odd and of the form 3m+2(m is a natural number) is equal to $(A)40\hspace{1 cm} (B)42\hspace{1 cm}(C)46\hspace{1cm}(D)52$ I could not identify factors,i think ...
5
votes
1answer
206 views

Wilson's Theorem - Why only for primes? [closed]

Why is it true that Wilson's Theorem only holds for prime numbers? I read a proof of it, and it did not seem to cater to that aspect of the theorem.
21
votes
4answers
1k views

Visualizing the factorial

Often in basic mathematics, we can visualize things very easily, which I believe helps understanding (instead of just working out a number theoretical proof). For example: $$(n+1)^2 - n^2 = (n+1) +n$$ ...
1
vote
1answer
54 views

Use stirlings approximation to prove inequality.

I have come across this statement in a text on finite elements. I can give you the reference if that will be useful. The text mentions that the inequality follows from Stirling's formula. I can't ...
-2
votes
1answer
33 views

A basic factorial question type [closed]

Hello could you show me a way that how to solve this kind of questions? a and b are natural numbers $60! = a6^b$ What is the biggest value of b?
0
votes
1answer
29 views

Help evaluating a partial sum with factorials and binomial coefficients

I come from a CS background and had to contend with a problem similar to this one. Essentially, I want a general-case estimate on how many rolls I'd have to make to land on the same number twice with ...
3
votes
4answers
74 views

Prove by induction: $\frac{1}{2!}+\frac{2}{3!}+\cdots+\frac{n}{(n+1)!}=\frac{n!-1}{n!}$

Prove $$\frac{1}{2!}+\frac{2}{3!}+\cdots+\frac{n}{(n+1)!}=\frac{n!-1}{n!}.$$ My problem with this is that it doesn't hold for the base case: $n=1$. This question is from the book "Abstract ...
6
votes
1answer
234 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 = ...
0
votes
0answers
29 views

How is zero factorial equals 1 [duplicate]

Simple question enough: Why is 0!=1 ? Had they chose it or proved it? Moreover is there any application of zero factorial?
0
votes
4answers
61 views

Find the Limit of the sequence $\frac{(2^n)(n!)}{(2n+1)!}$

Find the Limit of the sequence $\frac{(2^n)(n!)}{(2n+1)!}$ I'm not sure how to approach this problem. I tried the squeeze method, but could not figure it out.
0
votes
4answers
120 views

How many $0$s does the number $30!$ have? [duplicate]

I want to find out the number of $0$s in the number $30!$, what should I do? Is there any trick that would work for a general question of this type, like number of $0$s in $50!$ ?
16
votes
2answers
2k views

what is the remainder when $1!+2!+3!+4!+\cdots+45!$ is divided by 47?

Can any one please tell the approach or solve the question what is the remainder when $1!+2!+3!+4!+\cdots+45!$ is divided by $47$? I can solve remainder of $45!$ divided by $47$ using Wilson's ...
39
votes
3answers
695 views

Prove that there exist infinitely many integers $(n^{2015}+1)\mid n!$

I conjecture that there exist infinitely many integers $n$ such that $$(n^{2015}+1)\mid n!.$$ I have seen a simpler problem that there exist infinitely many integers $n$ such that $(n^2+1)\mid ...
0
votes
3answers
76 views

Induction: prove $2n^2 < 10\cdot n!$

Prove that $2n^2 < 10\cdot n!$, where $n$ is a positive integer My approach: $P(1)$ is true, and I'm trying to prove that $2(k+1)^2 < 10 (k+1)!$ Assume $2k^2 < 10\cdot k!$, and $2k^2 * ...
0
votes
1answer
49 views

Formula for reciprocal of a factorial

I was looking at some code here - https://www.codechef.com/viewsolution/6075682 when I came across this statement to calculate reciprocal of a factorial- ...
0
votes
2answers
52 views

A tricky diophantine equation with factorials

I am being unable to solve this diophantine equation. Does anyone have any suggestions. Let $n$ and $m$ both be non-negative integers. Find all solutions to $$n(nm - 2)! = (n!)^m$$ How would one ...
2
votes
3answers
102 views

what is remainder when $(((3!)^{5!})^{7!})^{9!…}$ is divided by 11

$$(((3!)^{5!})^{7!})^{9!...}$$ when divided by 11 what will be the reminder? Hint is appreciated Sorry I do not know how to start this problem, so I have not shown my efforts!
2
votes
0answers
46 views

Simplify ratio in factorial form [closed]

Is it possible to simplify the following ratio $$\frac{(\sum_{i=1}^{N}x_i)!}{(\prod_{i=1}^{N}x_i!)}$$ where $x_i\in\{0,1,2,..,M\}$
2
votes
2answers
183 views

How to sum factorials: $(n+1)! + n!$

How can the sum of factorials $(n+1)!+n!$ be simplified?
-1
votes
3answers
97 views
1
vote
0answers
32 views

Product of complex numbers $m+in$ with $0 < m,n \leq N$

I am trying to look for a generalization of Stirling's formula to complex numbers. In the integer case: $$ \log n! = \sum_{k = 1}^n \log k \approx \int_1^n \log x \, dx = n \log n - n$$ For the ...
1
vote
3answers
63 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 ...
3
votes
3answers
103 views

How many of the numbers in $A=\{1!,2!,…,2015!\}$ are square numbers?

Problem How many of the numbers in $A=\{1!,2!,...,2015!\}$ are square numbers? My thoughts I have no idea where to begin. I see no immediate connection between a factorial and a possible square. ...
2
votes
2answers
50 views

Wilson's theorem

According to Wilson's theorem, when p is prime (p-1)! mod p = -1 or p-1 What's the remainder in cases of (p-2)! mod p or ...