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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0
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4answers
35 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
110 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 ...
36
votes
3answers
618 views
+200

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
75 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
45 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
48 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
0answers
12 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
0answers
21 views

Sum of series involving factorials [closed]

$$ \sum\limits_{j=0}^{[\frac{n}{l}]}(-1)^{slj}\left( \begin{array}{c} n \\ lj \\ \end{array} \right)^s \frac{x^{kj}}{[(a)_{bj}]^s}, $$ where $l,s,n,k,a,b$ are natural numbers and x is ...
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
43 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
159 views

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

How can the sum of factorials $(n+1)!+n!$ be simplified?
-1
votes
3answers
95 views
1
vote
0answers
31 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
99 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
46 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 ...
3
votes
4answers
2k views

Calculate 2000! (mod 2003)

Calculate 2000! (mod 2003) This can easily be solved by programming but is there a way to solve it, possibly with knowledge about finite fields? (2003 is a prime number, so mod(2003) is a finite ...
2
votes
2answers
53 views

Is it possible to determine the number divisors of n! especially for large n?

I read this paper by P. Erdos, page 2. I didn't understand it. How do I determine the number divisors of $n!$ ? I'd like an example application, for example if I want to determine the number divisors ...
2
votes
1answer
74 views

Find the sum of the infinite series [1+(5/1!)+(8/2!)+(11/3!)+…]

Excluding the 1st term of the series 1, if we start from the 2nd term-($\frac{5}{1!}$), I can locate that the numerators are in A.P with common difference 3, & 1st term 5. Whereas the ...
0
votes
0answers
32 views

Math Factorials. Simplifying by distrubution. I am confused.

Say we are working with statistics and factorials. In the proof of ... $$\frac{n!}{r!(n-r)!} = \frac{n!}{(n-r)!(n-[n-r])!}$$ How is $(n-r)!(n-[n-r])!$ supposed to distribute to the simplified ...
2
votes
1answer
44 views

Number of digits in the number $N=(1.6 \times 10^{32})!$

I am trying to find the number of digits in $$N=(1.6 \times 10^{32})!$$ where ! denotes Factorial. I have no idea how to proceed, please help me.
3
votes
0answers
50 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 ...
1
vote
3answers
64 views

Help with proof of $(n+1)^n > n! 2^n$

I have already managed to prove it using induction and Bernoulli's inequality but I wonder if there is another way. My proof goes like this: (This is my first time using MathJax, so I apologize for ...
2
votes
3answers
217 views

What function approximates the growth in length of factorial?

The factorial function grows in length in digits faster and faster. For example, early on it is multiplied by tens, so grows one or two digits each time. Then in the hundreds it grows two or three ...
0
votes
1answer
66 views

Sum of primes at minimal $\gt t!$

$$2+3+5+17+97+599\cdots a_t \gt t!$$ What does that mean? Well it is a sum that follows specific rules. For one, the number of terms in the sequence is $t$. Similarly, $a_t$ represents the $t$'th ...
0
votes
0answers
37 views

Factorial-Like Symbol for polynomials?

Is there a symbol similar to the factorial for polynomials? Like if I say $4!=4\times 3\times 2\times 1$ What is the equivalent operation such that Operation: $(x)(4)=x^4+x^3+x^2+x$ Where $x$ is ...
0
votes
1answer
28 views

Comparison between exponential and factorial results

I'm developing an algorithm to compare if the result of $n!$ is bigger than $k^m$, but I have problems with big integers, then I need to know if there's some property that I can use to do this without ...
3
votes
3answers
69 views

Number of primes from $n!+1$ to $n!+n$

Why aren't there any primes between $n!+1$ and $n!+n$ for all $n>1$? This question was on AHSME 1969 #23, but the question is trivial because it's multiple choice. However, I have no idea how to ...
3
votes
9answers
164 views

Why is $0! = 1$ the same as $1! = 1$? [duplicate]

I want to ask why is $$0! = 1$$ the same as $$1! = 1.$$ As a student I was lost and when I tried to ask the question the teacher said this will be done in complex analysis. I know here I will ...
2
votes
2answers
65 views

Product of the first $N$ factorials

I'm trying to find a formula for the product of factorials: $$\prod _{n=1}^{N}n!=\; ?$$ Now using a kind of "brute force", I believe that I can prove that $$\prod _{n=1}^{N}n!=\prod ...
3
votes
3answers
100 views

Show $\frac{(2n)!}{n!\cdot 2^n}$ is an integer for $n$ greater than or equal to $0$

Show $$\frac{(2n)!}{n!\cdot 2^n}$$ is an integer for $n$ greater than or equal to $0$. Could anyone please help me with this proving? Thanks!
0
votes
2answers
38 views

Prove (or derive) the de Polignac formula for the prime decomposition of $n!$

I can't seem to find any papers published dedicated to show that the de Polignac formula has a rigorous derivation. From Wikipedia's entry for the formula: Let $n \geq 1$ be an integer. The prime ...
0
votes
1answer
35 views

Ratios between Factorial numbers and the sum of their factors

Let a factorial number be called $f!$. Let the sum of its factors be called $S(f!)$. Let the ratio between the two be “r”, such that $r=\frac{S}{f!}$. It is conjectured that: ...
7
votes
7answers
179 views

Prove $\lim\limits_{n\to\infty}\frac{1}{\sqrt[n]{n!}}=0$

I used $$(n!)^{\frac{1}{n}}=e^{\frac{1}{n}\ln(n!)}=e^{\lim\limits_{n\to\infty}\frac{1}{n}\ln(n!)}$$ Then using Stirling's approximation and L'Hospital's rule on ...
5
votes
0answers
48 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 ...
12
votes
4answers
17k 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 ...
13
votes
4answers
3k views

Factorial of infinity

So, I've read in this article that: $$\zeta'(0) = \log\sqrt\frac{1}{2\pi}$$ And that: $$e^{-\zeta'(0)} = 1\cdot2\cdot3\cdot\ldots\cdot\infty = \infty! = \sqrt{2\pi}$$ I found this result very ...
1
vote
0answers
22 views

Is there a function of, say, x and y that would take the first x factors in a factorial and return a xCy amounts of terms with y factors in each term?

What I'm basically looking for is described in the title. Here are some examples of what the function I'm looking for should do. Is there an existing function that does this? Even if not, are there ...
3
votes
1answer
36 views

Split Factorial of n

How can I split integers up to n into two groups such that the difference of the product of each group is as low as possible? Is there a way to optimize the selection for each group in order to ensure ...
2
votes
2answers
45 views

What is the logic/theorem/derivation behind finding the exponent of p in n! By [n/p] + [n/p^2] + [n/p^3] + …? [duplicate]

The exponent of prime number of 3 in 100! is 48. It means 100! is divisible by $3^48$ $$E_3(100!) = \left\lfloor\frac{100}3\right\rfloor + \left\lfloor\frac{100}{3^2}\right\rfloor + ...
4
votes
2answers
307 views

Exponent of Prime in a Factorial [duplicate]

I was just trying to work out the exponent for $7$ in the number $343!$. I think the right technique is $$\frac{343}{7}+\frac{343}{7^2}+\frac{343}{7^3}=57.$$ If this is right, can the technique be ...
2
votes
3answers
171 views

Exponent of $p$ in the prime factorization of $n!$

Exponent of $p$ in the prime factorization of $n!$ is given by $\large \sum \limits_{i=1}^{\lfloor\log_p n \rfloor } \left\lfloor \dfrac{n}{p^i}\right\rfloor $. Can this sum be simplified further to ...
7
votes
3answers
434 views

Some trouble with the induction

Prove, that for any positive integer $n \geqslant 2$ we have the inequality $$ \frac{ 4^n }{ n+1 } < \frac{ (2n)! }{ (n!)^2 }.$$ For $n=2$ the inequality is true. Directly just take and ...
1
vote
2answers
90 views

Any shortcut to calculate factorial of a number (Without calculator or n to 1)?

I've been searching the internet for quite a while now to find anything useful that could help me to figure out how to calculate factorial of a certain number without using calculator but no luck ...
-1
votes
1answer
18 views

Common factors /greatest common factors /least common factor

See I am baffling with the concept of above mentioned title I saw one question they wrote find common factors /gcf/lcm of 42,294,882 Please help me out in getting the concept Thanks
3
votes
1answer
104 views

Compute $(-1)^n\sum_{k=1}^n (-1)^k\frac{(k+n-1)!}{(k-1)!(k-1)!(n-k)!}$

Compute $(-1)^n\sum_{k=1}^n (-1)^k\frac{(k+n-1)!}{(k-1)!(k-1)!(n-k)!}$ Define $a_{k,m}=\frac{(-1)^{k+m}(n+k-1)!(n+m-1)!}{(k+m-1)[(k-1)!(m-1)!]^2(n-m)!(n-k)!}$ Compute ...
49
votes
1answer
2k 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 ...
0
votes
1answer
43 views

How do I prove the formula for multichoose?

In combinatorics, there is a formula "$n$ multichoose $k$", which is the way of making a multiset having $k$ elements choosing out of $n$ options. "$n$ multichoose $k$" is the same as "$(n+k-1)$ ...
1
vote
1answer
52 views

Approximating $\frac{(kn)!}{(n!)^k}$

Is there any approximations for the form $$\frac{(kn)!}{(n!)^k},$$ where $n$ and $k$ are positive integers? $n$ is not necessary much larger than $k$?