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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175
votes
6answers
9k views

How could we define the factorial of a matrix?

Suppose I have a square matrix $\mathsf{A}$ with $\det \mathsf{A}\neq 0$. How could we define the following operation? $$\mathsf{A}!$$ Maybe we could make some simple example, admitted it makes any ...
116
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7answers
16k views

Is there a function that grows faster than exponentially but slower than a factorial?

In big-O notation the complexity class $O(2^n)$ is named "exponential". The complexity class $O(n!)$ is named "factorial". I believe that $f(n) = O(2^n)$ and $g(n) = O(n!)$ means that $\dfrac{f(n)}{g(...
103
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1answer
5k 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 $...
91
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2answers
5k views

Prove elementarily that $\sqrt[n+1] {(n+1)!} - \sqrt[n] {n!}$ is strictly decreasing

Prove without calculus that the sequence $$L_{n}=\sqrt[n+1] {(n+1)!} - \sqrt[n] {n!}, \space n\in \mathbb N$$ is strictly decreasing.
90
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10answers
8k views

Is $0! = 1$ because there is only one way to do nothing?

The proof for $0!=1$ was already asked at here. My question, yet, is a bit apart from the original question. I'm asking whether actually $0!=1$ is true because there is only one way to do nothing or ...
89
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7answers
8k 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.
87
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13answers
93k views

Do factorials really grow faster than exponential functions? [closed]

Having trouble understanding this. Is there anyway to prove it?
69
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3answers
5k views

Why is $i! = 0.498015668 - 0.154949828i$?

While moving my laptop the other day, I ended up mashing the keyboard a little, and by pure chance managed to do a google search for i!. Curiously, Google's ...
63
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18answers
13k views

Prove $0! = 1$ from first principles

How can I prove from first principles that $0!$ is equal to $1$?
63
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16answers
50k views

Prove that $\lim \limits_{n \to \infty} \frac{x^n}{n!} = 0$, $x \in \Bbb R$.

Why is $$\lim_{n \to \infty} \frac{2^n}{n!}=0\text{ ?}$$ Can we generalize it to any exponent $x \in \Bbb R$? This is to say, is $$\lim_{n \to \infty} \frac{x^n}{n!}=0\text{ ?}$$ This is being ...
58
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3answers
982 views

Is $\lfloor n!/e\rfloor$ always even for $n\in\mathbb N$?

I checked several thousand natural numbers and observed that $\lfloor n!/e\rfloor$ seems to always be an even number. Is it indeed true for all $n\in\mathbb N$? How can we prove it? Are there any ...
57
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10answers
12k 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 = \sqrt{...
55
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16answers
7k 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 ...
55
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4answers
126k views

Factorial, but with addition [duplicate]

Is there a notation for addition form of factorial? $$5! = 5\times4\times3\times2\times1$$ That's pretty obvious. But I'm wondering what I'd need to use to describe $$5+4+3+2+1$$ like the ...
55
votes
3answers
58k views

$\sum k! = 1! +2! +3! + \cdots + n!$ ,is there a generic formula for this?

I came across a question where I needed to find the sum of the factorials of the first $n$ numbers. So I was wondering if there is any generic formula for this? Like there is a generic formula for ...
55
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5answers
18k views

Highest power of a prime $p$ dividing $N!$

How does one find the highest power of a prime $p$ that divides $N!$ and other related products? Related question: How many zeros are there at the end of $N!$? This is being done to reduce abstract ...
50
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12answers
8k views

$\lim\limits_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite [closed]

How do I prove that $ \displaystyle\lim_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite?
50
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5answers
4k views

Inequality from Chapter 5 of the book *How to Think Like a Mathematician*

This is from the book How to think like a Mathematician, How can I prove the inequality $$\sqrt[\large 7]{7!} < \sqrt[\large 8]{8!}$$ without complicated calculus? I tried and finally obtained ...
46
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3answers
5k views

$n!+1$ being a perfect square

One observes that \begin{equation*} 4!+1 =25=5^{2},~5!+1=121=11^{2} \end{equation*} is a perfect square. Similarly for $n=7$ also we see that $n!+1$ is a perfect square. So one can ask the truth of ...
44
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5answers
97k views

How to find the factorial of a fraction?

From what I know, the factorial function is defined as follows: $$n! = n(n-1)(n-2) \cdots(3)(2)(1)$$ And $0! = 1$. However, this page seems to be saying that you can take the factorial of a fraction,...
44
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3answers
2k views

How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes? [duplicate]

Possible Duplicate: Highest power of a prime $p$ dividing $N!$ How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes?
43
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3answers
7k views

Is $n! + 1$ often a prime?

Related to another question (If $n = 51! +1$, then find number of primes among $n+1,n+2,\ldots, n+50$), I wonder: How often is $n!+1$ a prime? There is a related OEIS sequence A002981, however, ...
42
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5answers
130k views

Derivative of a factorial

What is ${\partial\over \partial x_i}(x_i !)$ where $x_i$ is a discrete variable? Do you consider $(x_i!)=(x_i)(x_i-1)...1$ and do product rule on each term, or something else? Thanks.
41
votes
2answers
18k views

How many zeroes are in 100!

One common math puzzle I've seen around asks for how many zeros are in the product of "100!" Usually, the solution everyone gives goes something like try to match pairs of 5s and 2s that factor out ...
40
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3answers
3k views

How is Gosper's approximation to factorial derived?

Discussing Stirling's approximation, Wolfram Mathworld article mentions a modification of it due to Gosper: instead of the usual $$n!\approx n^ne^{-n}\sqrt{2n\pi},$$ we have a tiny addition of $\...
38
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5answers
6k views

Showing that $\frac{\sqrt[n]{n!}}{n}$ $\rightarrow \frac{1}{e}$ [duplicate]

Show:$$\lim_{n\to\infty}\frac{\sqrt[n]{n!}}{n}= \frac{1}{e}$$ So I can expand the numerator by geometric mean. Letting $C_{n}=\left(\ln(a_{1})+...+\ln(a_{n})\right)/n$. Let the numerator be called $...
37
votes
8answers
3k views

What's the value of $\sum\limits_{k=1}^{\infty}\frac{k^2}{k!}$?

For some series, it is easy to say whether it is convergent or not by the "convergence test", e.g., ratio test. However, it is nontrivial to calculate the value of the sum when the series converges. ...
37
votes
4answers
2k views

Which is bigger: $9^{9^{9^{9^{9^{9^{9^{9^{9^{9}}}}}}}}}$ or $9!!!!!!!!!$?

In my classes I sometimes have a contest concerning who can write the largest number in ten symbols. It almost never comes up, but I'm torn between two "best" answers: a stack of ten 9's (exponents) ...
37
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3answers
1k 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 $(n^2+1)\mid n!$. ...
36
votes
12answers
15k views

Division of Factorials

I have a partition of a positive integer $(p)$. How can I prove that the factorial of $p$ can always be divided by the product of the factorials of the parts? As a quick example $\frac{9!}{(2!3!4!)} =...
36
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1answer
1k views

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 ...
34
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10answers
12k views

What is the purpose of Stirling's approximation to a factorial?

Stirling approximation to a factorial is $$ n! \sim \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n. $$ I wonder what benefit can be got from it? From computational perspective (I admit I don't know ...
34
votes
4answers
2k views

A closed form of $\sum_{k=1}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right)$

I am looking for a closed form of the following series \begin{equation} \mathcal{I}=\sum_{k=1}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right) \end{equation} I have no idea how to ...
34
votes
1answer
954 views

A series problem by Knuth

I came across the following problem, known as Knuth's Series which originally was an American Mathematical Monthly problem. Prove that $$\sum_{n=1}^\infty \left(\frac{n^n}{n!e^n}-\frac{1}{\sqrt{2\...
31
votes
8answers
2k views

Prove that $(mn)!$ is divisible by $(n!)\cdot(m!)^n$

Let $m$ be a positive integer and $n$ a nonnegative integer. Prove that $$(n!)\cdot(m!)^n|(mn)!$$ I can prove it using Legendre's Formula, but I have to use the lemma that $$ \dfrac{\...
30
votes
10answers
3k views

Without using a calculator, is $\sqrt[8]{8!}$ or $\sqrt[9]{9!}$ greater?

Which is greater between $$\sqrt[8]{8!}$$ and $$\sqrt[9]{9!}$$? I want to know if my proof is correct... \begin{align} \sqrt[8]{8!} &< \sqrt[9]{9!} \\ (8!)^{(1/8)} &< (9!)^{(1/9)} \\...
30
votes
7answers
32k views

The product of $n$ consecutive integers is divisible by $n$ factorial

How can we prove that the product of $n$ consecutive integers is divisible by $n$ factorial? Note: In this subsequent question and the comments here the OP has clarified that he seeks a proof that "...
30
votes
3answers
650 views

Constructing $\mathbb N$ from the set of factorials

Let S be the set $\{0!, 1!, 2!, \ldots\}$. Is it possible to construct any positive integer using only addition, subtraction and multiplication, and using any element in S at most once? For example: $...
29
votes
6answers
3k views

$m!n! < (m+n)!$ Proof?

Prove that if $m$ and $n$ are positive integers then $m!n! < (m+n)!$ Given hint: $m!= 1\times 2\times 3\times\cdots\times m$ and $1<m+1, 2<m+2, \ldots , n<m+n$ It looks simple but I'm ...
28
votes
2answers
12k views

Is there a way to solve for an unknown in a factorial?

I don't want to do this through trial and error, and the best way I have found so far was to start dividing from 1. $n! = \text {a really big number}$ Ex. $n! = 9999999$ Is there a way to ...
28
votes
2answers
1k views

Did I derive a new form of the gamma function?

I wish to extend the factorial to non-integer arguments in a unique way, given the following conditions: $n!=n(n-1)!$ $1!=1$ To anyone interested in viewing the final form before reading the whole ...
27
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2answers
4k views

Is $\sqrt{n!}$ a natural number?

I'm new here (on Mathematics Stack Exchange). Also, I'm a 10th grade student not a math expert. So, my question is whether, $$\sqrt {n!} $$ comes in the set of the Natural Numbers. There were ...
27
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1answer
1k views

Repeated Factorials and Repeated Square Rooting

I was talking with friends about silly questions involving what numbers you can get using only a single digit "3" and unary operations. We eventually conjectured that using only factorials and square ...
26
votes
7answers
4k views

Find $n$, where its factorial is a product of factorials

I need to solve $3! \cdot 5! \cdot 7! = n!$ for $n$. I have tried simplifying as follows: $$\begin{array}{} 3! \cdot 5 \cdot 4 \cdot 3! \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3! &= n! \\ (3!)^3 \...
26
votes
3answers
6k views

Limit of nth root of n! [duplicate]

I am asked to determine if a series converges or not: $\displaystyle\sum\limits_{n=1}^{\infty} (2^n)n!/(n^n)$ So I'm using the nth root test and came up with $\lim_{n \to {\infty}}(2/n)*(\sqrt[n]{n!})...
26
votes
1answer
371 views

Numbers that are clearly NOT a Square

Although I have never studied math very seriously, I have heard of Brocard's Problem, which asks for integer solutions for the following Diophantine Equation:$$n!+1=m^2$$ The only solutions are ...
25
votes
2answers
2k views

Repeatedly taking differences on a polynomial yields the factorial of its degree?

Consider a function such that it takes in polynomial function and creates an array of its outputs and then using that array creates another new array by calculating the absolute difference between the ...
25
votes
4answers
3k views

Solutions for x!/y!=(y+1)!

I was watching a video recently, and I saw how 10*9*8*7 was equal to 7*6*5*4*3*2*1, or to make it clearer, 10!/6!=7!. I was wondering if there were any other solutions, so I checked the web, to find ...
24
votes
6answers
14k views

If $n\ne 4$ is composite, then $n$ divides $(n-1)!$.

I have a proof and need some feedback. It seems really obvious that the statement is true but it is always the obvious ones that are a little trickier to prove. So I would appreciate any feedback. ...
24
votes
5answers
8k views

Can n! be a perfect square when n is an integer greater than 1?

Can n! be a perfect square when n is an integer greater than 1? (But is it possible, to prove without Bertrand's postulate. Because bertrands postulate is quite a strong result.)