Questions tagged [prime-factorization]
For questions about factoring elements of rings into primes, or about the specific case of factoring natural numbers into primes.
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GCD and LCM using Prime Factorization
I saw in a book that we can find the LCM and GCD of three numbers using prime factorization . That was really cool :) I'll explain what i saw and will let you know my doubt in the end!
Three numbers ...
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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 $1$.
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What software can calculate the order of $b \mod p$, where $p$ is a large prime?
I wasn't sure where to ask this, but Mathematics seems better than StackOverflow or Programmers.
I have no background whatsoever in number theory, and I need to find software that can calculate the ...
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Does $a \mid bc$ imply $\frac{a}{(a,b)} \mid c$?
If $a \mid bc$, then does $\frac{a}{(a,b)} \mid c$? I doubt anybody here is industrious enough to show this via a diagram, but who knows.
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Let $f(n)$ be the number of prime factors of the positive integer $n$. Find $\lim_{n\to \infty}\frac{f(n)} n$
Let $f(n)$ be the number of prime factors of the positive integer $n$. Find $\displaystyle \lim_{n\to \infty}\frac{f(n)} n$.
I suspect it's equal to $0$, but how can I show this? Thank you.
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Efficiency in factoring lists of consecutive numbers
Suppose I'm looking at prime factorizations of numbers in the vicinity of this one:
$$
1354 = 2 \times 677
$$
The smallest prime appears here, and the next prime after that does not.
Going one step ...
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prime divisor of $3n+2$ proof
I have to prove that any number of the form $3n+2$ has a prime factor of the form $3m+2$. Ive started the proof
I tried saying by the division algorithm the prime factor is either the form 3m,3m+1,3m+...
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Verifying prime factorization equivalence class
I define a relation on $\Bbb N$ as follows:
$x \sim y \Longleftrightarrow \ \exists \ j,k \in \Bbb Z$ s.t. $x \mid y^j \ \wedge \ y \mid x^k$
I have shown that $\sim$ is an equivalence relation by ...
- 4,467
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How to use fundamental theorem of arithmetic to conclude that $\gcd(a^k,b^n)=1$ for all $k, n \in$ N whenever $a,b \in$ N with $\gcd(a,b)=1$?
How to use fundamental theorem of arithmetic to conclude that $\gcd(a^k,b^n)=1$ for all $k, n \in$ N whenever $a,b \in$ N with $\gcd(a,b)=1$?
Fundamental theorem of arithmetic: Each number $n\geq 2$ ...
user2723
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About the factors of the product of prime numbers
If a number is a product of unique prime numbers, are the factors of this number the used unique prime numbers ONLY? Example: 6 = 2 x 3, 15 = 3 x 5. But I don't know for large numbers. I will be using ...
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Mean of highest exponent in prime factorization of all integers ≥ 2
For any natural number $n > 1$, define $E(n)$,to be the highest exponent to
which a prime divides it. For instance, $E(12)=E(36)=2$. Show that $$\lim_{N \to \infty} \frac{1}{N} \sum\limits_{n=2}^{N} E(...
anonymous