Tagged Questions
1
vote
1answer
42 views
Understanding the upper and lower bounds of the error estimate in Stirling's Approximation
Based on the Wikipedia article on Stirling Approximation:
$n! = \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n e^{\lambda_n}$
where $\frac{1}{12n+1} < \lambda_n < \frac{1}{12n}$
How would this ...
2
votes
2answers
143 views
Analyzing the lower bound of a logarithm of factorials using Stirling's Approximation
I am trying to get the lower bound for:
$f(x) = \ln(\lfloor\frac{x}{4}\rfloor!) - \ln(\lfloor\frac{x}{5}\rfloor!) -\ln(\lfloor\frac{x}{20}\rfloor!) - 2(1.03883)(\sqrt{\frac{x}{4}}) - ...
3
votes
1answer
54 views
Reasoning about the Chebyshev functions: How does one check an upper bound based on the second Chebyshev function?
In Ramanujan's proof of Bertrand's Postulate, Ramanujan states:
$\log([x]!) - 2\log([\frac{1}{2}x]!) \le \psi(x) - \psi(\frac{1}{2}x) + \psi(\frac{1}{3}x)$
where:
$\vartheta(x) = \sum_{p \le x} ...
8
votes
0answers
216 views
Understanding Ramanujan's approach in his proof of Bertrand's Postulate
I've been reading through Ramanujan's proof of Betrand's Postulate and I'm not clear why he didn't state his proof in terms of $\varphi(2x) - \varphi(x)$
What would be wrong with this approach for ...
1
vote
2answers
33 views
Constrictions on A.P with factorials.
There are five numbers $(a_1,a_2,a_3,a_4,a_5)$, such that they are in Arithmetic Progression.
Given that $a_1$ and $a_2$ are factorials, is there a possibility that either $a_4$ OR $a_5$ is a ...
3
votes
2answers
67 views
Factorials and Arithmetic Progression.
Are there sets of factorials $(a_1!,a_2!,a_3!,\dots,a_n!)$, such that they exist in Arithmetic progression.
$n$ is a natural number
I don't see any such examples(Except for $n=2$). And I don't see ...
12
votes
1answer
118 views
Solving $n!+m!+k^2=n!m!$ for positive integers $n,m,k$
I have been running in circles with this for a while now.
It seems that the only solution is $(n,m,k)=(2,3,2)$ but I don't know how to prove it.
Things I have noticed: WLOG $n\geq m$ we see that ...
2
votes
3answers
80 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! = ...
7
votes
2answers
200 views
Number of zero digits in factorials
Here is a riddle someone has been asked in a job interview: How many zero digits are there in $100!$?
Well, I found the first $24$ quite fast by counting how many times five divides $100!$ ($5$ ...
10
votes
2answers
192 views
Prove quotient of factorials is integral
If $n$ is an integer $\gt 0$, prove
$$\frac{(30n)!n!}{(15n)!(10n)!(6n)!}$$
is also an integer. I understand that a general approach is proving that the power of any prime factor is greater in the ...
2
votes
1answer
90 views
Need help with Factorial Sums! [duplicate]
Possible Duplicate:
How to prove that the number 1!+2!+3!+…+n! is never square?
Show that the sum $$\sum_{k=1}^nk!\neq m^2$$for any integer $m$, for $n\geq4$.
2
votes
3answers
148 views
When is a factorial of a number equal to its triangular number?
Consider the set of all natural numbers $n$ for which the following proposition is true.
$$\sum_{k=1}^{n} k = \prod_{k=1}^{n} k$$
Here's an example:
$$\sum_{k=1}^{3}k = 1+2+3 = 6 = 1\cdot 2\cdot ...
2
votes
0answers
123 views
Generating all positive integers from three operations
This question arose on sci.math, but since almost all the competent mathematicians there have migrated here, I thought I'd give the question a wider audience. Starting with an integer $t>2$, ...
3
votes
0answers
95 views
Prove: $\frac{(2px)!}{((px)!)^2}\equiv\frac{(2x)!}{((x)!)^2}\pmod{p^2}$
How can I prove the following, where $p$ is a prime and $x$ a positive integer?
$$\dfrac{(2px)!}{((px)!)^2}\equiv\dfrac{(2x)!}{((x)!)^2}\pmod{p^2}$$
I'm not sure if it is actually true, but I tested ...
5
votes
2answers
177 views
Find the remainder when $ 12!^{14!} +1 $ is divided by $13$
Find the remainder when $ 12!^{14!} +1 $ is divided by $13$
I faced this problem in one of my recent exam. It is reminiscent of Wilson's theorem. So, I was convinced that $12! \equiv -1 \pmod {13} $ ...
2
votes
1answer
124 views
If a function is a product of factorials of polynomials, how can I prove that this representation is unique?
If a function is a product of factorials of polynomials, how can I prove that this representation is unique?
Specifically, let $F(x) = \prod_i{P_i(x)!^{a_i}}$, $F:\mathbb{N} \rightarrow \mathbb{Q}$, ...
10
votes
4answers
768 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 ...
1
vote
1answer
118 views
Can someone explain to me the relationship between primorials and factorials and how that relation can be used to compute large factorials?
What I am trying to figure out is a way to compute large factorials, !1000000.
For what it's worth luschny's computer algorithms do a very good job of it.
7
votes
1answer
2k views
Last non Zero digit of a Factorial
I ran into a cool trick for last non zero digit of a factorial. This is actually a recurrent relation which states that:
If $D(N)$ denotes the last non zero digit of factorial, then
...
3
votes
1answer
256 views
Smallest positive integer
Let S(m) is the sum of the factorials of the digits of integer m.
I try to find the smallest positive integer n with S(n)=111.
My answer is 12334444. Is it right?
4
votes
1answer
304 views
On the factorial equations $A! B! =C!$ and $A!B!C! = D!$
I was playing around with hypergeometric probabilities when I wound myself calculating the binomial coefficient $\binom{10}{3}$. I used the definition, and calculating in my head, I simplified to this ...
3
votes
2answers
700 views
Derive a formula to find the number of trailing zeroes in $n!$ [duplicate]
Possible Duplicate:
How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes?
I know that I have to find the number of factors of $5$'s, $25$'s, ...
7
votes
2answers
214 views
Finding all the numbers that fit $x! + y! = z!$
I have the formula $x! + y! = z!$ and I'm looking for positive integers that make it true. Upon inspection it seems that x = y = 1 and z = 2 is the only solution. The problem is how to show it.
...
7
votes
2answers
149 views
Closed form for $(p-n)!\pmod{p}$ where $p$ is prime
Does $(p-n)!\pmod{p}$ have a closed form for any $n>2$ when $p$ is prime?
$(p-0)!=0 \pmod{p}$
$(p-1)!=-1\pmod{p}$
$(p-2)!=1\pmod{p}$
0
votes
1answer
151 views
$90!$ when divided by $n$, gives an odd number. What can be the minimum and the maximum values of $n$?
$90!$ when divided by $n$, gives an odd number. How could we find the minimum and the maximum values of $n$?
I am not sure how to approach this one, any ideas?
15
votes
6answers
2k views
prove that $(2n)!/(n!)^2$ is even if $n$ is a positive integer
Prove that $(2n)!/(n!)^2$ is even if $n$ is a positive integer. For clarity: the denominator is the only part being squared.
My thought process: The numerator is the product of the first n even ...
1
vote
1answer
303 views
What is the theoretical upper bound of factorion numbers?
Recently I read about factorion numbers. I understood that there are only 4 factorion numbers, but what is the theoretical range in which they can be? Is it $[0, +\infty]$ or a smaller upper range? ...
8
votes
2answers
256 views
Number of zeros not possible in $n!$ [duplicate]
Possible Duplicate:
How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes?
The number of zeros which are not possible at the end of the $n!$ is:
...
37
votes
3answers
1k 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?
