# 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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### 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 ...
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### 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 ...
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### 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 $$(n(n+1)+1)!+1\neq m^2$$...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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)
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### 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 ...
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### 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 $$y \cdot y!= \frac{q}{x \cdot x!}$$ ...
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### 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 ...
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### 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 ...