All Questions
20
questions
4
votes
0
answers
84
views
Finite many primes for every positive integer $b$?
Consider the function $$f(a,b):=\sum_{j=0}^a (bj)!=1+b!+(2b)!+\cdots +(ab)!$$
Given a positive integer $b$ , are there always only a finite number of positive integers $a$ such that $f(a,b)$ is prime ...
- 85.1k
0
votes
1
answer
63
views
If n is a positive integer that is four digits long and is relatively prime to 100!, why must n be prime?
Suppose there is some positive integer n that is four digits long and is relatively prime to 100! (meaning n and 100! have no common factors other than 1). n must be prime, but why?
100! is a ...
2
votes
1
answer
2k
views
Find the power of a prime in the prime factorization of a large factorial
I have been trying to work through the following exercise:
Find the power of $5$ in the prime factorization of $2020!$.
So far I have worked out that the prime factorization of $2020$ is $2^2 \cdot 5^...
- 143
0
votes
3
answers
63
views
how (a!)/(b!) = (b + 1)×(b + 2)×⋯×(a − 1)×a [closed]
I was solving a problem in which i need to figure out the prime factorization of $\frac{a!}{b!}$ and i did that by computing (a!) and then (b!) by looping ((1 to a) & (1 to b)) and then derived n ...
- 111
3
votes
1
answer
76
views
Are there infinitely many composiphiles?
I came up with this today: we say a positive integer $k$ is a composiphile if there exists no positive integer $n \leq k$ such that
$$\frac{k!}{n} + 1 \text{ is prime.}$$
My question: are there ...
58
votes
2
answers
8k
views
Can I search for factors of $\ (11!)!+11!+1\ $ efficiently?
Is the number $$(11!)!+11!+1$$ a prime number ?
I do not expect that a probable-prime-test is feasible, but if someone actually wants to let it run, this would of course be very nice. The main hope ...
- 85.1k
1
vote
0
answers
45
views
Number dividing the factorial of a lowest prime
I would like to ask a probably simple question but I am not sure how to make it rigorous.
Let $p$ be a prime and $n$ a natural number whose lowest prime factor is $p$. If a natural number $x$ divides ...
- 727
1
vote
3
answers
356
views
Power of prime in prime factorization of a factorial.
Please advise on how to arrive at solution for determining the power of 17 in the prime factorization of 2890!
Also, is there a short-cut?
So far I know:
Prime factorization of 2890 = 2 x 5 x 17^2
...
- 131
-1
votes
4
answers
1k
views
Last non zero digit in 20! [duplicate]
So I have a question where it says to find the last non zero digit of $20!$
I proceeded in the following way:
Found the prime factorization of $20!$ by calculation the greatest powers of $2,3,5,7,11,...
- 1,454
1
vote
2
answers
367
views
Logarithm of factorial equal to sum of logarithm of primes
Let $N$ a positive integer. Denote $\mathcal{P}$ the set of prime numbers. I have to show that
\begin{align}
\log(N!) = \sum_{p^{\nu}\leq N \\ p\in \mathcal{P}} \left\lfloor\dfrac{N}{p^{\nu}}\right\...
- 866
3
votes
0
answers
47
views
How to find the prime factorization of a very large number. [duplicate]
I want to know if there are any tricks or shortcuts to write the factorial of a large number, like $20!,$ as the product of its prime factors.
For example, $5!= 5 \times 3 \times 2^3$
- 79
6
votes
1
answer
2k
views
Find smallest positive integer $n$ such that $n!$ is divisible by $p^k$ ($p =$ positive prime, $k =$ positive integer)
I have to find smallest positive integer $n$ in such way that $n!$ is divisible by $p^k$ ($p$ is always positive prime and $k$ is always positive integer).
$p$ and $k$ are given, $n$ is (obviously ...
user506862
4
votes
1
answer
56
views
Reasoning about a sequence of consecutive integers and factorials with hope of relating factorials to primorials
I am looking for someone to either point out a mistake or help me to improve the argument in terms of clarity, conciseness, and more standard mathematical argument.
Let $x$ be an integer such that $x,...
- 10.5k
1
vote
2
answers
483
views
Prime factorization of factorials
Is there a way given a sequence of naturals $a_1, a_2, ..., a_k$ to determine whether $c=n!$ for some number $n$ where
$$ c = 2^{a_1} 3^{a_2}5^{a_3}7^{a_4}...$$ (2,3,5,7,... - primes)
- 593
3
votes
3
answers
389
views
The ultimate formula to factor them all.
Context
I am working on Integer factorization problem, I found a formula for factoring numbers, and I need your help to simplify it. First I will explain how I get there and then I present the ...
- 1,450