Linked Questions

2
votes
1answer
244 views

$n! =$ the product of consecutive integers. [duplicate]

Can $n!$ be the product of $k$ consecutive integers for $k > 1$? (Not including the degenerate cases such as when $k = 2$, then $1\cdot2 = 2!$ and $2\cdot 3 = 3!$, and so on.) I am asking not for $...
3
votes
1answer
139 views

Finding natural numbers $a, b, c$ such that $a!=b!\times c!$ [duplicate]

Yesterday, when I was playing with numbers, I was surprised to know the following relation: $$10!=6!\times 7!.$$ Then, I've been looking for the other solutions, but I'm facing difficulty. Then, ...
4
votes
0answers
101 views

How to find other nontrivial solutions of $a!b!=c!$? [duplicate]

I know only one nontrivial solution of this equation: $6!\cdot 7!=10!$. There is also a series of trivial solutions: $n!(n!-1)!=(n!)!,\ \forall n\in\mathbb{N}$. So my question is how to find any other ...
4
votes
0answers
75 views

Find $a,b,c$ such that $a!\times b!\times c!=d!$ [duplicate]

I have to find $a,b,c \in \mathbb{N}$ such that-$a!\times b!\times c!=d!$ Answer given in my book is $3!\times5!\times7!=10!$(But it is written that other answers are also possible). What is a ...
3
votes
0answers
70 views

Factorials and prime factorial factors [duplicate]

Is it just a coincidence that 10! = 3! · 5! · 7! and 6! = 3! · 5! or are factorials somehow related to primes in terms of prime factorial factors?
1
vote
0answers
50 views

Factorial Triples [duplicate]

Lately I've come up with an interesting math problem to work at. I've been trying to find "factorial triples", or triples of numbers that satisfy $$A!B!=C!$$ with $A,B \ne 1$. By playing around, I've ...
0
votes
0answers
32 views

The diophantine equation $a! = b! \times c!$ [duplicate]

One infinite family of solutions to the equation $a! = b! \times c!$ has $a = s!$ so we have $(s!)! = (s!-1)! \times s!$ but I'm hard pressed to find manually another type of solution apart from $10! =...
2
votes
4answers
2k views

Factorial expressed in terms of two other factorials

Can the factorial of $N$ always be expressed by the sum(addition and subtraction) or the product of two other factorials? Do there always exist integer $A$ and $B$ such that $N! = A! + B!$, or $N! = ...
14
votes
3answers
212 views

$n!$ as product of consecutive numbers

Let $n$ be a positive integer. In how many ways can one write $n!$ as a product of consecutive integers? For example: $4!=1\times2\times3\times4=2\times3\times4$. Here, $2$ possibilities exist. $...
11
votes
2answers
186 views

When is a binomial coefficient a factorial, i.e. when is $\binom{m}{j} = n!$ for some $n,m,j$?

As stated in the title: when is a binomial coefficient a factorial, i.e. when is $\binom{m}{j} = n!$ for some $m,j,n$? I was thinking about this problem a couple of days ago because in all my years of ...
9
votes
0answers
174 views

products of factorials

Let $F = \{n! \mid n > 1\}$. Let $S_1$ be the set of integers which can be expressed as a product of one or more (not necessarily distinct) elements of $F$. Let $S_2$ be the set of integers which ...
3
votes
1answer
102 views

Four fractions of certain factorials give another factorial

Let $n>0$ and $s_n=\sum_{k=1}^n k$. I looked at the expressions $\displaystyle\frac{s_n!}{(s_n-n)!}$ and found that the fraction is another factorial for $k=1,2,3,4$, i.e. $$\frac{1!}{0!}=1=1!\;,...
2
votes
1answer
67 views

Integer solutions of $\Gamma(a)\Gamma(b) = \Gamma(c)$

Are there infinitely many nontrivial integer solutions $(a,b,c)$ of $$\Gamma(a)\Gamma(b)=\Gamma(c)\hspace{10mm}?$$ This seems like it may have been asked before but I didn't find an earlier ...
4
votes
1answer
116 views

Diophantine equation involving factorials

$$x!+y=y^3$$ $$y=\sqrt[3]{x!+\sqrt[3]{x!+\sqrt[3]{x!+\cdots}}}$$ The only integer solutions to these identities that I have found are: $$3!+2=2^3$$ $$4!+3=3^3$$ $$5!+5=5^3$$ $$6!+9=9^3$$ I ...
3
votes
0answers
94 views

Find all possible solutions!

Find solutions for $$^nP_r=s!$$ For $(n,r,s)\in \mathbb{N}$ I could find some trivial solutions $(6,3,5)~,~(1,1,1)$ etc.