Tagged Questions
4
votes
1answer
63 views
Last non zero digit of $n!$ [duplicate]
What is the last non zero digit of $100!$?
Is there a method to do the same for $n!$?
All I know is that we can find the number of zeroes at the end using a certain formula.However I guess that's of ...
4
votes
0answers
55 views
Trying to show that $\ln(x!) - \ln(\lfloor\frac{x}{2}\rfloor!) - \ln(\lfloor\frac{x}{3}\rfloor!) - \ln(\lfloor\frac{x}{6}\rfloor!) \ge \psi(x)$
I've been told that the approach below will not work.
I would be interested if someone could help me to understand what will go wrong.
Let:
$$\psi(x) = \sum\limits_{p^k \le x} \ln p$$
So that (see ...
3
votes
2answers
83 views
A question about prime factorization of n!
Prove that for any integer $K$, There exists a natural number $N$ so that in the prime factorization of $N!$ we can find at least $K$ prime numbers which their powers are exactly 1.
4
votes
0answers
78 views
Generalizing Jitsuro Nagura's result: my resulting upper bound for the second chebyshev function is too low. What am I doing wrong?
I've been reading through Jitsuro Nagura's classic proof that there is a prime between $x$ and $\frac{6x}{5}$ and it seems to me that it should be possible to improve on his upper bound for the second ...
0
votes
0answers
58 views
Using Stirling's Formula to approximate a difference of logarithms of factorials in the same way as Jitsuro Nagura.
In Jitsuro Nagura's classic proof of a prime existing between $x$ and $\frac{6x}{5}$, he uses Stirling's formula to show that:
$$T\left(x\right) - T\left(\frac{x}{2}\right) - ...
0
votes
0answers
42 views
Do these inequalities regarding the gamma function and factorials work?
I am seeing if it is possible to generalize lemma 1 in Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$.
In a previous question, I asked whether the following inequality is ...
2
votes
0answers
55 views
Trying to generalize an inequality from Jitsuro Nagura: Does this work?
I am investigating the generality of Lemma 1 in Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$.
In Lemma 1, Nagura establishes when $n > 1$, $x \ge 1$:
...
0
votes
1answer
68 views
Proving that a specific gamma function is a guaranteed lower bound for a factorial function
In reviewing Ramanujan's proof of Bertrand's postulate, Ramanujan observes that:
$$\ln\Gamma(x) - 2\ln\Gamma(\frac{x+1}{2}) \le \ln(\lfloor{x}\rfloor!) - 2\ln(\lfloor\frac{x}{2}\rfloor!)$$
I have ...
0
votes
1answer
37 views
Is it possible to generalize Ramanujan's lower bound for factorials when $\{\frac{x}{b_2}\} + \{\frac{x}{b_3}\} < 1$?
This is the second attempt at a proof. My first attempt had a flaw in its logic.
After reviewing the mistake in logic, I believe that with a revised logic, the argument can be saved.
The revision ...
1
vote
0answers
28 views
Is it possible to generalize Ramanujan's lower bound for factorials when $\{\frac{x}{b_2}\} + \{\frac{x}{b_3}\} \ge 1$?
This is the second attempt at a proof. My first attempt had a flaw in its logic.
After reviewing the mistake in logic, I believe that with a revised logic, the argument can be saved.
The revision ...
8
votes
1answer
205 views
Is it possible to generalize Ramanujan's lower bound for factorials that he used in his proof of Bertrand's Postulate?
I am starting to feel more confident in my understanding of Ramanujan's proof of Bertrand's postulate. I hope that I am not getting overconfident.
In particular, Ramanujan's does the following ...
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} ...
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 ...
5
votes
1answer
171 views
Question about Ramanujan's proof of Bertrand's Postulate
I am reviewing Ramanujan's proof of Bertrand's Postulate which can be found here.
At step #7, he writes: "But it is easy to see that..."
$\log\Gamma(x) - 2\log\Gamma\left(\frac{1}{2}x + ...
3
votes
1answer
71 views
Understanding a very elementary property of factorials
I've seen this stated in a few places.
If $$\vartheta(x) = \sum_{p\le{x}} \log (p) \qquad \psi(x) = \sum_{m=1}^{\infty}\vartheta\left(\sqrt[m]{x}\right)$$ Then $$\log(x!) = \sum_{m=1}^{\infty} ...
3
votes
2answers
68 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
119 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 ...
3
votes
3answers
81 views
Find all values of $n$ greater than or equal to 1 for which $n! + (n + 1)! + (n + 2)!$ is equal to a perfect square.
Not sure where to get started on this on. I started listing numbers for n starting at 1 but the numbers get very big very fast and I cannot find a pattern. Is there a better way of doing this or ...
7
votes
5answers
290 views
Find all solutions of the equation $x! + y! = z!$ [duplicate]
Not sure where to start with this one. Do we look at two cases where $x<y$ and where $x>y$ and then show that the smaller number will have the same values of the greater? What do you think?
4
votes
1answer
74 views
What is the analytic continuation of a multifactorial?
The $\Gamma$ function is the analytic continuation of the factorial function. Is there a similar analog for multifactorials?
I am particularly interested in the double factorial. All Google has ...
7
votes
2answers
204 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 ...
29
votes
2answers
412 views
Constructing $\mathbb N$ from the set of factorials
Let S be the set $\{0!, 1!, 2!, \ldots\}$. Is it possible to construct any positive integer using only addition, subtraction and multiplication, and using any element in S at most once? For example:
...
0
votes
0answers
102 views
How to find the last non-zero digit in ${^n\!P_k} $?
What is the procedure of finding the last non-zero element in ${^n\!P_k}$?
3
votes
2answers
93 views
Summation of a binomial style sum, equal to gamma function?
I am trying to prove the following for a very, very long time:
$$\sum_{k=0}^j \frac{ (-1)^k}{k! (j-k)!} \frac{1}{2k+1} = \frac{1}{2} \frac{\sqrt{\pi}}{\Gamma(3/2 + j)}$$
or, equivalently
...
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
...
13
votes
2answers
327 views
On the Limit of Stirling's Approximation
I have recently proven the following curious identity: For real $x \geqslant 1$,
\begin{align}
\lfloor x \rfloor! = x^{\lfloor x \rfloor} e^{1-x} e^{\int_{1}^{x} \text{frac}(t)/t \ dt}
\end{align}
...
1
vote
0answers
99 views
Efficient factorion search in arbitrary base
A factorion in base $N$ is a natural number equal to the sum of the factorials of its digits in base $N$. So, the decimal factorions are:
$1 = 1!$
$2 = 2!$
$145 = 1! + 4! + 5!$
$40585 = 4! + 0! + 5! ...
2
votes
1answer
82 views
Understanding bounds on factorions
I am trying to understand the upper bound on factorions (in base $10$). The Wikipedia page says:
"If $n$ is a natural number of $d$ digits that is a factorion, then
$10^{d − 1} \le n \le 9!d$. ...
21
votes
1answer
463 views
are all $n!$ ($n>3$) the difference of two squares?
For the small values of n I have been able to check, it seems that for $n>3$, there exist whole numbers $x,y$ s.t. $n! = x^2 - y^2$. For example ..
$4! = 5^2 - 1^2$
$5! = 11^2 - 1^2$
$6! = 27^2 ...
3
votes
3answers
332 views
Finding all positive integer solutions to $(x!)(y!) = x!+y!+z!$
The equation is $(x!)(y!) = x!+y!+z! $
where $x,y,z$ are natural numbers.
How to find out them all?
3
votes
0answers
152 views
How to find $\beta$ and $\alpha$?
$\mathbb{P}$ is the prime numbers set.
$p \in \mathbb{P}$
$a,b,c \in \mathbb{N}$
$n=a p^b+c$ where
$c= n\bmod p$
$b$ is the highest power of $p$ who divides $n-c$
How to find $\beta$ where ...
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 ...
29
votes
1answer
1k 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 ...
8
votes
4answers
685 views
How many consecutive composite integers follow k!+1?
I wrote a program for myself in Mathematica to generate the answer for the first 300, which was really easy, but I can't find a pattern. The results are here. This is a problem in Underwood Dudley's ...
2
votes
1answer
425 views
Factorial of a non-integer number
My TI-83 calculator doesnt allow me to do this, but using Windows calculator, I can compute the factorial of say 5.8. What does this mean and how does it work?
