All Questions
19
questions
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 ...
- 85.1k
1
vote
0
answers
50
views
Is the value of the sum squarefree for every $n\ne 27\ $?
For a positive integer $\ n\ $ , define $$f(n)=|\sum_{j=1}^n (-1)^j\cdot j!|=n!-(n-1)!+(n-2)!-(n-3)!\pm \cdots$$
Is $\ f(n)\ $ squarefree except for $\ n=27\ $ in which case $\ 127^2\ $ is a factor ?
...
- 85.1k
6
votes
1
answer
98
views
Conjecture about prime factors of a special sum
For a positive integer $\ n\ $ , define $$f(n)=|\sum_{j=1}^n (-1)^j\cdot j!|=n!-(n-1)!+(n-2)!-(n-3)!\pm \cdots$$
I want to prove the
Conjecture : Every prime factor $\ p\ $ of $\ f(n)\ $ must satisfy ...
- 85.1k
0
votes
1
answer
102
views
What is $x$, if $1! + 2! + 3! + \cdots + (x-1)! + x! = k^2$ and $k$ is an integer?
What is $x$, if $1! + 2! + 3! + \cdots + (x-1)! + x! = k^2$ and $k$ is an integer?
Using trial and error, it is obvious that for $x < 4$, the given equation has solutions only for $x = 1,k = \pm1$ ...
- 725
3
votes
2
answers
690
views
How many values of $n$ are there for which $n!$ ends in $1998$ zeros?
How many values of $n$ are there for which $n!$ ends in $1998$ zeros?
My Attempt:
Number of zeros at end of $n!$ is
$$\left\lfloor \frac{n}{5}\right\rfloor+\left\lfloor\frac{n}{5^2}\right\rfloor+\...
- 9,599
1
vote
1
answer
91
views
Prove that $\sum_{k=1}^{\infty }\left \lfloor \frac{n}{p^k} \right \rfloor=\frac{n-S_n}{p-1}$
If $p$ is a prime number, $n$ is a natural number, and $S_n$ is the sum of the digits of $n$ when expressed in base $p$.
$$\text{Prove that }\sum_{k=1}^{\infty }\left \lfloor \frac{n}{p^k} \right \...
- 5,450
2
votes
1
answer
52
views
I don't understand one thing in a solution of a number theory task
I'm going through a solution of a number theory task and I don't understand one thing.
Here's the task
$d_1,d_2,...,d_n$ are all natural factors of $10!$. Find the following sum
This sum can be ...
- 192
4
votes
4
answers
569
views
What are the trailing number of the zeroes in the given integer
Problem Statement:- The number of zeroes at the end of the integer
$$100!-101!+\ldots-109!+110!$$
I am having a bit of a trouble in thinking how do I proceed. A little push in the right direction ...
- 2,758
0
votes
1
answer
47
views
Is there any formula to compute $\sum\limits_{i=1}^{i=k}\lfloor{\frac{n}{5^i}\rfloor}$?
This is the formula we use to calculate the number of trailing zeros in $n!$ but we have to add the individual elements of the sum to get up with an answer. Is there any way we can bypass that?
- 2,411
5
votes
1
answer
554
views
What is the value of $\sum_{k=1}^{n}k!$? [duplicate]
What is the sum of all the factorials starting from 1 to n? Is there any generalized formula for such summation?
- 2,411
2
votes
2
answers
371
views
Finite summation including binomial coefficients and double factorials
I came across the following summation:
$$
\sum_{k=0}^n\frac{(-1)^k(2k)!!}{(2k+1)!!}\dbinom{n}{k}\,\,\,\,(n\in\mathbb{N}).
$$
$\tbinom{n}{k}$ are binomial coefficients, $n!/k!(n-k)!$.
Mathematica told ...
- 35
7
votes
2
answers
1k
views
proof - Show that $1! +2! +3!+\cdots+n!$ is a perfect power if and only if $n=3$
Show that $1! +2! +3!+\cdots+n!$ is a perfect power if and only if $n =3$
For $n=3$, $1!+2!+3!=9=3^2$. I also feel that the word 'power' makes it a whole lot hard to prove. How do we prove this? What ...
- 4,042
2
votes
2
answers
2k
views
Verify If Sum of Factorials is Divisible by Integer
I am working on preparing for JEE and was working on this math problem.
We have the sum, $$\sum_{n=1}^{120}n!=1!+2!+3!+\ldots+120!$$
Now I am given the question, which says that what happens when ...
- 23
4
votes
4
answers
1k
views
What is the units digit of $\sum\limits_{n=1}^{1337} (n!)^4$?
What is the units digit of $\sum\limits_{n=1}^{1337}(n!)^4$ ?
I find 9 but I am not sure.
- 97
0
votes
2
answers
65
views
Factorial Summation Problem [duplicate]
$$\sum_{j=0}^n j\cdot j!$$
I got $(n+1)!-1$ as the answer but I'm not sure if that's right or how I even got to that answer exactly. (my paper is a mess of random work and I can't make it out). Can ...
- 21