0
votes
1answer
76 views

Given a prime p and an integer N, find the number of integers n such that 1≤n≤N and order(n!) is divisible by p

We are given a prime number $\leq 10^{18}$ and an integer N $(\leq N\leq 10^{18})$ how to find the number of integers lying in the range $1\leq n\leq N$ for which the order(n!) is a multiple of p? ...
35
votes
3answers
3k views

Is $n! + 1$ often a prime?

Related to this question, I wonder how often $n!+1$ is a prime? There is a related OEIS sequence A002981, however, nothing is said if the sequence is finite or not... or anything in that sense...
10
votes
4answers
844 views

If $n = 51! +1$, Then find no of primes among $n+1,n+2,\ldots, n+50$

If $n = 51! +1$, Then find no of primes among $n+1,n+2,\ldots, n+50$ Really speaking, I don't have any clue ...
1
vote
2answers
41 views

prove that $N$ is divisible by $1,2,\ldots,k$ which $k+1$ is the lowest prime number after $N$

Suppose $n$ is a natural number ($n\ge 5$) and $k+1$ is the lowest prime number that is greater than $n$ prove that $A_i \mid n!$ which $A_i$ are these numbers: $1,2,\ldots,k$
1
vote
2answers
63 views

Number of primes in $[30! + 2, 30! + 30]$

How to find number of primes numbers $\pi(x)$ in $[30! + 2$ , $30! + 30]$, where $n!$ is defined as: $$n!= n(n-1)(n-2)\cdots3\times2\times1$$ Using Fermat's Theorem: $130=1\mod31$, (since $31 \in ...
0
votes
1answer
40 views

Summation of a curious series-repeated division by primes

I am interested in knowing if there is some closed form/formula for the following series: ...
1
vote
2answers
268 views

Factorials and Prime Factors

I need to write a program to input a number and output it's factorial in the form: $4!=(2^3)(3^1)$ $5!=(2^3)(3^1)(5^1)$ I'm now having trouble trying to figure out how could I take a number and get ...
3
votes
2answers
619 views

How many perfect squares divide 1!2!3!4!5!6!7!8!9!

What I naturally did was to find the prime factorisation of the product of factorials which is $ 2^{30}3^{13}5^5 7^3 $. Clearly there is 15 unique perfect squares that divide $2^{30}$, 6 unique ...
4
votes
2answers
87 views

Show that $p!$ and $(p - 1)! - 1$ are relatively prime

If $p$ is prime number, with $p>3$ Show that $p!$ and $(p - 1)! - 1$ are relatively prime. I tried $\text{gcd}\;(p!,(p-1)!-1)=d\Longrightarrow d\mid p!$ e $d\mid(p-1)!-1$ having ...
1
vote
1answer
34 views

show that if $m,n\in\mathbb{N}$ are such that $(m,n)=1$, then $\frac{(m+n-1)!}{m!n!}\in\mathbb{N}.$

show that if $m,n\in\mathbb{N}$ are such that $(m,n)=1$, then $$\frac{(m+n-1)!}{m!n!}\in\mathbb{N}.$$ I have a theorem (shown in my text) that says $$\fbox{If $a_1,...,a_m\in\mathbb{N}^*$ then ...
7
votes
2answers
84 views

Find the greatest power of $104$ which divides $10000!$

Find the greatest power of $104$ which divides $10000!$ I thought $$104=2^3\cdot13$$ so I have to find $n$ such that $$(2^3\cdot13)^n\mid 10000!$$ Obviously, we can see that there are fewer ...
4
votes
1answer
57 views

Congruences with prime number and factorial

Prove that if $p\equiv 1 \pmod{4}$ is a prime number and $$x\equiv \pm \left(\frac{p-1}{2}\right)! \pmod{p}$$ then $x^2\equiv -1 \pmod{p}$ I think Wilson's theorem will come in handy here, used ...
1
vote
2answers
30 views

Does $n=n^2 - (n!\;\bmod n^2)\implies\text{isPrime}(n) = \text{True}$?

With integers $n$, of such form that $$n=n^2 - (n!\mod n^2)$$ Is $n$ always a prime number?
3
votes
1answer
52 views

Why does $(k-2)!-k \left\lfloor \frac{k!}{(k-1) k^2}\right\rfloor = 1,\;k\ge2\;\implies\;\text{isPrime}(k)$

Let $k$ be a integer such that $k\ge2$ Why does $$(k-2)!-k \left\lfloor \frac{k!}{(k-1) k^2}\right\rfloor = 1$$ only when $k$ is prime? Example: $$\pi(n) = \sum _{k=4}^n \left((k-2)!-k \left\lfloor ...
8
votes
0answers
429 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 ...
5
votes
1answer
379 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 + ...
5
votes
2answers
444 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} $ ...
3
votes
0answers
194 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 ...
8
votes
4answers
861 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 ...