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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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3 votes
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Permanent divisor in highly composite numbers

I was wondering whether it is true that if a particular divisor (be it prime or composite) appears for the first time in the sequence of highly composite numbers (HCNs), would it still be present for ...
BarbaraKwarc's user avatar
2 votes
0 answers
44 views

How to define double factorial for non positive integers?

I studied double factorial which known for natural number $$ n!!=n(n-2)!! , 1!!=0!!=1$$ So we have for $n\in N$ $$ (2n)!!=2^n n! , (2n+1)!!=\frac{(2n+1)!}{2^n n!}$$ but I found on Math-World formula ...
Faoler's user avatar
  • 1,607
-3 votes
1 answer
250 views
+50

Like Ramanujan : Strange limit and a remarkable behavior .

Problem: Let $f_a\left(x\right)=a^{3-\sqrt{1+2x!\sqrt{1+3x!!\sqrt{1+4x!!!\sqrt{\cdot\cdot\cdot\sqrt{1+(k-1)x!\cdots!\sqrt{1+kx!!\cdots!}}}}}}}, g_a(x)=\left(\frac{f_a\left(x\right)}{f_a\left(0\right)}\...
Ranger-of-trente-deux-glands's user avatar
-1 votes
1 answer
53 views

Is this expression valid? [closed]

I encountered this expression in twitter,but I am not sure if it is correct. $$\lim\limits_{x \to \infty} \frac{x!}{x^x}$$ The thing is, the factorial is only defined for positive integer numbers. I ...
CAPA's user avatar
  • 13
0 votes
0 answers
54 views

Applying convex multiplicative functions to Brocard's Problem

Brocard's problem asks if there are integer solutions to $n! = (x-1)(x+1)$ other than the cases of $n =$ $4$, $5$, $7$. Knowing the only shared divisor of the factors on the right is 2, would it be ...
PiMaster's user avatar
2 votes
2 answers
45 views

Determining the witnesses (constants $C_0$ and $k_0$) when showing $c^n \in O(n!)$ ($c > 1$)

I'm having a hard time finding the constants/witnesses $C_0$ and $k_0$ that show $c^n \in O(n!)$. That is $|c^n| \leq C_0|n!|$ for $n > k_0$ ($c > 1$). To be clear, I understand we can prove the ...
Bob Marley's user avatar
0 votes
0 answers
19 views

Extracting coefficients from falling factorial form in Mathematica

I have a polynomial in falling factorial form, i. e. $-x+7(-1+x)x+6(-2+x)(-1+x)x+(-3+x)(-2+x)(-1+x)x$. Now i want to extract the "outermost" coefficients $\{-1,7,6,1\}$. I have this code to ...
mhighwood's user avatar
  • 141
0 votes
0 answers
51 views

Divisors of $x^2-1$ in Brocard's Problem

In this post, I was curious if the divisor bound could be improved for the product of two consecutive even numbers. It seems most likely it cannot by much. How could the upper bound of $\sigma_0(x^2 - ...
PiMaster's user avatar
2 votes
1 answer
85 views

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

I have tried substituting $(2n)! \sim \sqrt{4\pi n} \left(\frac{2n}{e}\right)^{2n}$ from Stirling's approximation: $$\begin{align*} \lim_{n\to\infty} \frac{(-1)^{n}n^{2n}}{(2n)!} &= \lim_{n\to\...
Anonymous Account's user avatar
0 votes
0 answers
31 views

The sum of the signed stirling numbers times the factorials

The sum of interest is the following... $$ \sum^{n}_{k=1} (k-1)!s(n,k) $$ Where the $s(n,k)$ are the signed Stirling number of the first kind. The sum is very close to other identities involving the ...
Aidan R.S.'s user avatar
-1 votes
1 answer
79 views

Without calculating the value of $34!$, determine the digits $a$ and $b$ in the representation 34! = 295232799039a041408476186096435b0000000

Without calculating the value of $34!$, determine the digits $a$ and $b$ in the representation $$34! = 295232799039a041408476186096435b0000000,$$ assuming that the remaining digits are correctly ...
user avatar
-1 votes
0 answers
36 views

Find the remainder when $(1!1) + (2!2) + (3!3) +...+(286!*286)$ is divided by $2 ^ 4 * 5 ^ 3$. [duplicate]

I did this $1×1! = 2! - 1!, 2×2! = 3! - 2!$ and so on then by telescoping I get $287! - 1!$ so as $287!$ is divisible by $2000$ so got the remainder as $1999$, but the answer is given $08$. So please ...
Shubhank agase's user avatar
4 votes
0 answers
159 views

Can we efficiently compute $a!\mod b$?

It is well known that we easily can compute , say , $2^a\mod b$ for large integers $a,b$. We can use the repeated square method which gives a fast result even if $a,b$ have , say , $50$ decimal digits....
Peter's user avatar
  • 85k
0 votes
0 answers
102 views

Find the sum of infinite series: $\sum\limits_{n=1}^{\infty} \frac{(\beta \, n)^n}{n!} $

I am looking for the sum of an infinite series: $$ \sum_{n=1}^{\infty} \frac{(\beta \, n)^n}{n!} $$ where $\beta$ is a constant between $0$ and $1$. I know that if $\beta < 1/e$, the series ...
Weihua Gu's user avatar
2 votes
3 answers
218 views

Wrong way to find $a$ if remainder of $25!/(23!-a)$ is $3600$?

The question is: $a$ is a single-digit number, and the remainder from $25!/(23!-a)$ is $3600$. What is $a$? (Source: A problem in Turkish university acceptance exam (AYT).) Some guy on Reddit claimed ...
cekos's user avatar
  • 31
0 votes
1 answer
37 views

Inequality involving exponentials and factorials

Let $b>1$ (a base), $n\ge 2$ and $1\le k\le n$. I would like to know for which $k$ the inequalities $$ \frac{1}{n!}b^{n-1}\le \frac{b^k-1}{(n-k)!}\le\frac{1}{(n-1)!}(b^n-1)(b-1). $$ hold or at ...
mathemagician99's user avatar
3 votes
0 answers
86 views

Infinite summation of a falling factorial divided by a power

I'm trying to find the result of this summation $$ S(r,m)=\sum_{n=m}^\infty \frac{(n)_m}{n^r} $$ where $(r,m\in\mathbb{N})$, $(r\geq3)$, $(1\leq m \leq r-2)$ and $(n)_m=\frac{n!}{(n-m)!}$ is a falling ...
Max Pierini's user avatar
2 votes
1 answer
39 views

Prove $\lim_{n\to\infty}\binom{n}{k}\left(\frac{\mu}{n}\right)^{k}\left(1-\frac{\mu}{n}\right)^{n-k}=\frac{\mu^k}{e^\mu\cdot{k!}}$

The problem is to prove the following equality: $$\lim_{n\to\infty}\binom{n}{k}\left(\frac{\mu}{n}\right)^{k}\left(1-\frac{\mu}{n}\right)^{n-k}=\frac{\mu^k}{e^\mu\cdot{k!}}$$ This is what I have ...
Aidan Hyde's user avatar
1 vote
0 answers
91 views

Double factorial primes

Let's say that a number is a double factorial prime if it is a prime and there is a number $n$ such that it is equal to $(n!)! \pm 1$ (here $(n!)!$ is the factorial of the factorial, not the usual $n!!...
Weier's user avatar
  • 785
10 votes
2 answers
167 views

Faster approach: Find the smallest integer $a$ such that $P = a\cdot1!\cdot2!\cdot3!\cdot4!\cdot5!....\cdot18!$ is a perfect square.

Find the smallest integer $a$ such that $P = a\cdot1!\cdot2!\cdot3!\cdot4!\cdot5!....\cdot18!$ is a perfect square. This is a multiple-choice problem for a time-tight exam so I need to be as fast as ...
ten_to_tenth's user avatar
  • 1,426
2 votes
1 answer
119 views

What is the value of $\sum_{n=2}^∞ 1/(n! - 1)$?

I am trying to work out the value of $\sum_{n=2}^∞ 1/(n! - 1)$ as an exercise. The series is very similar to $\exp(1)$, but I am struggling with it. I tried to differentiate $f(x) = \sum_{n=2}^∞ x^n/(...
user avatar
-1 votes
2 answers
124 views

Does this identity exist? $1=(-1)!+(-2)!$ [closed]

Does this identity exist? $$ 1 = \left(-1\right)! + \left(-2\right)! $$ where "$!$" is the factorial function. Because $$ n! - \left(n - 1\right)! = \left(n - 2\right)! \quad\implies\quad n\...
Lucien Jaccon's user avatar
-2 votes
1 answer
496 views

Rewriting in a special case that Brocard's problem have only finite primitive solution i.e Brown's numbers [closed]

I recently found a possible rewirting in the affirmative of the most famous Brocard problem or Ramanujan-Brocard problem: Problem : Let $n>3$ and $m>1$ be integers then $$(n(n+1)+1)!+1\neq m^2$$...
Ranger-of-trente-deux-glands's user avatar
-1 votes
1 answer
111 views

Why we can find $(-\frac{3}{2})!$ but not $(-1)!$?

I understand that the gamma function is an extension of the factorial function to complex numbers and that it is not defined for non positive integer values. For example, we cannot find the value of $\...
Prince Yadav's user avatar
1 vote
2 answers
68 views

Find constant for asymptotic equivalence (Stirling's formula)

Find constant $A$ such that $C_{n^2}^{n} \sim A\frac{n^{2n}}{n!}$ Writing $C_{n^2}^{n}$ using Stirling's formula gives $C_{n^2}^{n} \sim \frac{n^{2n}}{\sqrt{2\pi}\sqrt{n-1}} \sim \frac{n^{2n}}{\sqrt{2 ...
John Doe's user avatar
2 votes
2 answers
99 views

Prove that $\exp \left(\dfrac{-2 \sum_{n=0}^{K-1} \frac{2^n}{n!}}{ \sum_{n=0}^K \frac{2^n}{n!}}\right) \sum_{n=0}^{K-1} \dfrac{2^n}{n!} -1 \geq 0$

I want to show the following inequality: $$\exp \left(\dfrac{-2 \displaystyle \sum_{n=0}^{K-1} \frac{2^n}{n!}}{\displaystyle \sum_{n=0}^K \frac{2^n}{n!}}\right) \displaystyle \sum_{n=0}^{K-1} \dfrac{2^...
user avatar
0 votes
0 answers
41 views

Efficient computation of factorial in finite field

What is the state of the art for fast computation of the factorial function $n!$ and more generally of $\prod_{1\leq x\leq n} (x-q)$ (rising/falling factorial) in a finite field? I found just one ...
Jim's user avatar
  • 538
1 vote
1 answer
104 views

What explains these numerators being the same?

In figuring out the number of terms of different order in the adjugate from this question, I stumbled upon this fact: The numerators for the alternating sum of fractional factorials (https://oeis.org/...
julianiacoponi's user avatar
1 vote
0 answers
81 views

Closed form for $ \sum_{a_1=0}^\infty~\sum_{a_2=0}^\infty~\cdots~\sum_{a_n=0}^\infty \dfrac1{(a_1!+a_2!+\ldots+a_n!)} $ [closed]

After reading this post and the general solution for that case, I wonder if there is a closed form for the general solution for this sum: $ \sum_{a_1=0}^\infty~\sum_{a_2=0}^\infty~\cdots~\sum_{a_n=0}^\...
user967210's user avatar
1 vote
1 answer
36 views

Could there be a function $f$ that satisfies $g(s)=f^{(s)}(x)^s|_{x=1/e}=s!$?

I was thinking about the factorial function today and I wondered: Is there a function $f$ that satisfies $g(s)=f^{(s)}(x)^s|_{x=1/e}=s!$ where the notation is the $s$th derivative of $f$ to the $s$th ...
zeta space's user avatar
0 votes
0 answers
49 views

Last $4$ nonzero digits of $n!$ in base fourteen

Question: What are the last $4$ nonzero digits of $2025!$ in base fourteen? Note: This is not a contest/homework question. I know how to find the number of trailing zeroes in base fourteen, so $2025!$ ...
Thirdy Yabata's user avatar
0 votes
0 answers
14 views

Interesting behavior of a function in a 2D XY plane

Consider the function y =x^n/n! plotted for ( n = 2 ) to ( 20 ). Upon examination, it becomes evident that for certain larger values of ( n ) and ( y ), the gap between the graphs corresponding to two ...
Hridoy Ranjan Kalita's user avatar
1 vote
1 answer
72 views

Every natural number has a unique representation in the factorial number system

I want to prove the following proposition: Let $n$ be a natural number. Then there is a unique representation of the form $n=c_1\cdot1!+c_2 \cdot 2!+...+c_k \cdot k!$. Hint: Use the identity: $\sum_{k=...
NTc5's user avatar
  • 609
5 votes
1 answer
231 views

Closed form for $\Gamma(a-x)$ where $a \in (0,1]$.

Now asked on MO here. I wonder if there is a closed form for $ \Gamma(a-x)$. And by closed form here I mean a finite combinations of elementary functions, powers of $\Gamma(a)$ and powers of $\Gamma(...
pie's user avatar
  • 6,416
5 votes
2 answers
156 views

$\sum_{n=1}^{\infty} \frac{n 4^n}{(2 n-1)^2(4 n+1)(4 n+3)} \frac{\binom{2 n}{n}}{\binom{4 n}{2 n}}=\frac{4(1+\sqrt{2})}{225}$ [closed]

I want to show that $\sum_{n=1}^{\infty} \frac{n 4^n}{(2 n-1)^2(4 n+1)(4 n+3)} \frac{\binom{2 n}{n}}{\binom{4 n}{2 n}}=\frac{4(1+\sqrt{2})}{225}.$ I tried manipulating the terms in the sum, but that ...
Sam's user avatar
  • 3,350
5 votes
2 answers
148 views

Irrationality of numbers that are the sum of reciprocal factorials, like $e.$

I wish to prove that for any infinite $A\subset \mathbb{N},\ \displaystyle\sum_{k\in A} \frac{1}{k!}$ is irrational, in the same manner that Rudin proves $e$ is irrational. I should mention the ...
Adam Rubinson's user avatar
3 votes
1 answer
94 views

Are there any prime numbers $\geq 5$ which are not a factor of some $n!-1,\ $ where $n\geq 2$?

For each $n\in\mathbb{N},$ let $S_n$ be the set of prime factors of $n! + 1$. By Wilson's theorem, we have $\ p\mid (p-1)!+1\ $ for every prime $p.$ Therefore, $\displaystyle\bigcup_{n=1}^{\infty} S_n ...
Adam Rubinson's user avatar
0 votes
0 answers
33 views

Proof that sequence of factorials is a Benford sequence without Weyl criterion

I know of a short proof that shows that the proportion of powers of $2$ with first digit '$1$' is $\log_{10}(2)$. It involes analysing the intervals of positive numbers that start with $1$: $[1,1), [...
Xaver Wallenstein's user avatar
4 votes
1 answer
178 views

Do iterated factorials or iterated exponents grow faster?

This is purely out of curiosity and I'm not quite at the point in calculus where I know how to prove either for myself... Given $$n^{n^{n^{n^n}}}$$ and $$(((n!)!)!)!$$ As n approaches infinity, which ...
æ æ's user avatar
  • 86
0 votes
0 answers
74 views

Is it possible to find a closed form for $i!$? [duplicate]

I am curious is there a closed form for $i!$? I tried to search for any closed form for this but I didn't find any. $$z! := \lim_{n \to \infty } n^z \prod_{k=1}^n \frac {k}{z+k}$$ $$i! =\lim_{n \to \...
Mathematics enjoyer's user avatar
1 vote
0 answers
46 views

Determine if factorial of exponent is bigger than exponent of factorial

For natural values $a$, $b$ determine if $(a^b)! > a^{b!}$. My thoughts : since logarithm is strictly monotonic $a > b \iff \ln(a) > \ln(b)$, let's consider $\ln(a^b)! - \ln(a^{b!})$. Using ...
Vitaliy Volovyk's user avatar
0 votes
1 answer
46 views

Falling and rising factorial series identity

$$ \sum_{n=0}^\infty\prod_{m=1}^n\frac{x-m+1}{km} = \sum_{n=0}^\infty\prod_{m=1}^n\frac{x+m-1}{(k+1)m} $$ I noticed this identity that relates the falling and rising factorials using a power series. ...
stackshifter's user avatar
3 votes
0 answers
75 views

Can we show that $\frac{\sum_{j=1}^n j^2\cdot j!}{99}$ generates only finite many primes?

Define $$f(n):=\frac{\sum_{j=1}^n j^2\cdot j!}{99}$$ Is $f(n)$ prime for only finite many positive integers $n\ge 10$ ? Approach : If we find a prime number $q>11$ with $q\mid f(q-1)$ , then for ...
Peter's user avatar
  • 85k
26 votes
1 answer
534 views

Prime factor wanted of the huge number $\sum_{j=1}^{10} j!^{j!}$

What is the smallest prime factor of $$\sum_{j=1}^{10} j!^{j!}$$ ? Trial : This number has $23\ 804\ 069$ digits , so if it were prime it would be a record prime. I do not think however that this ...
Peter's user avatar
  • 85k
-3 votes
3 answers
74 views

How does $(k+1)!(k+2)(k+1)$ simplify to $(k+2)!(k+1)$

If $$n!=n(n-1)!$$ then $$(k+1)!= (k+1)k(k-1)!$$ and $$(k+2)!$$ would be $$(k+2)(k+1)k(k-1)!$$ or $$(k+2)(k+1)!$$ but what does the extra (k+1) do to make it (k+2)!(k+1)
Pranav Borse's user avatar
2 votes
2 answers
75 views

Ways To Order Matches - Six Nations

There are $n$ teams in a sports tournament, and each team has to play every other team, and all teams have to play every weekend over $n-1$ weekends. For example, rugby six nations $n=6$ pretty much ...
Colm Bhandal's user avatar
  • 4,719
2 votes
1 answer
120 views

Is it possible to re-arrange LHS for $x$ or $y$ in the equation $x(y!)(x!)y=q$?

I was wondering if it's possible to rearrange for $y$ or $x$ in the following equation containing factorials? $$x(y!)(x!)y = q $$ I try to solve for $y$ as $$y \cdot y!= \frac{q}{x \cdot x!}$$ ...
Factorial_123's user avatar
19 votes
2 answers
551 views

Is there a prime number $p$ dividing $1+2!^2+3!^2+\cdots (p-1)!^2$?

Is there a prime number $p$ with $p \mid \sum_{j=1}^{p-1} j!^2$? I checked the primes upto $600\ 000$ without finding a solution. Heuristic : If we can assume that the probability that $p$ is a ...
Peter's user avatar
  • 85k
2 votes
1 answer
115 views

Limit $\lim_{n\to +\infty}{\frac{\left(n!\right)^2\cdot4^n\cdot n}{\left(2n\right)!}}$ [duplicate]

I am currently trying to calculate the following limit of sequence: $$\lim_{n\to +\infty}{\frac{\left(n!\right)^2\cdot4^n\cdot n}{\left(2n\right)!}}$$ I need it to prove that a series diverges, but I ...
Vito Palmieri's user avatar
-3 votes
1 answer
54 views

How to set equations containing factorials? [closed]

I recently encountered a problem involving the construction of equations involving factorials or combinatorial numbers. I recall reading somewhere (although I cannot recall the reference) that this ...
Dimitris's user avatar
  • 797

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