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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questions with no upvoted or accepted answers
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Can we rule out $\ p\mid t\ $ , if $\ p\mid \phi_t(n)\ $?
Let $\ \phi_m(x)\ $ denote the $\ m\ $-th cyclotomic polynomial.
For $\ n\ge 2\ $ define $\ t:=\phi_n(n)\ $
Must all the prime factors $\ p\ $ of $\ \phi_t(n)\ $ satisfy $\ p\equiv 1\mod t\ $ ?
It ...
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Conjecture on the sum of prime factors
$\text{Notations}$
Let $\pi(n)$ be the prime counting function.
Let denote $\alpha(n)$ the sum of the prime factors of $n$. (i.e. In other words, if $$n=p_1^{x_1}p_2^{x_2}...p_m^{x_m}$$ then $\alpha(n)...
user799688
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On the least number of factors $\sigma(q^{e_q})$ to get the least multiple of $\operatorname{rad}(n)$, on assumption that $n$ is an odd perfect number
I don't know if my question can be answered easily from your reasonings and knowledeges of the theory of odd perfect numbers. I wondered about it yesterday. Now this post is cross-posted on ...
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Is it possible to generate a composite number with no information about its factors?
Are there functions or algorithms which can generate integers which are necessarily composite, yet not yield any information about what factors it has?
For example, $f(x):=x^2-1$ is not what I'm ...
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Finding 'glitch' primes
Consider numbers of the form
$$
p=\underbrace{b\,\cdots\, b}_{n\,b\text{'s}}\,a\,\underbrace{b\,\cdots\, b}_{n\,b\text{'s}},
$$
where $0\leq a\leq 9$ and $1\leq b\leq 9$. In other words,
$$
p=(a - b)...
- 3,435
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Can I efficiently enumerate all numbers in a range that have a prime factor in another given range?
Suppose $a<b$ are positive integers. The object is to determine all the numbers $x\in [a,b]$ having a prime factor in the range $[c,d]$ efficiently (that is without factoring all the numbers in the ...
- 85.1k
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Review of proof attempt of Bertrand-Chebyshev Theorem
In the first line of Chapter Three of my graduate level text from which I am working from, the author declares the following divisibility relation to be the basis of Chebyshev theorem,
but has also ...
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What is the smallest $k$ , such that $44^{44}+k$ has the desired property?
I search the smallest positive integer $k$, such that $44^{44}+k$ splits into three distinct prime factors each having $25$ decimal digits.
The $21$-digit number $k=621725397145122340237$ does the ...
- 85.1k
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How to compare numbers by prime factorizations?
Let's say I have two natural numbers $n$ and $m$, both of which are exceeding large, and I want to determine which is smaller. Thankfully, I happen to know both of their prime factorizations, but they'...
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What is the largest number smaller than 100 such that the sum of its divisors is larger than twice the number itself?
What is the largest number smaller than 100 such that the sum of its divisors is larger than twice the number itself?
After doing some guess and check, I found that $36$ had quite a few factors, and ...
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Largest factored composite Cunningham number?
A number of the form $b^{n}\pm 1$ is a Cunningham number,
with non-square integer $b>1$ and integer $n>2$. The Fermat and Mersenne numbers are Cunningham numbers. The Cunningham Project ...
- 21.4k
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Is every primorial number squarefree?
Is every primorial number ( a number of the form $p$#$\pm 1$ ) squarefree ?
According to my calculation, there is no prime $q\le 270,000$, such that
$q^2$ can be a divisor of $p$#$-1$ or $p$#$+1$. ...
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If $x$ and $y$ are positive integers then $\frac{(2x)!(2y)!}{x!y!(x+y)!}$ is an integer
If $x$ and $y$ are positive integers then $\frac{(2x)!(2y)!}{x!y!(x+y)!}$ is an integer
I have to show that the proposition above is true for any $x,y\in\mathbb{Z^+}$ by means of Legendre's formula.....
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Are there unique factorizations for weyl algebras?
I just read about Weyl algebras, and they sound like neat little toys that are similar in a number of ways to polynomials. However, it's curious to me that they are non-commutative, and I was ...
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Integer Factorization via Trigonometry
Nearly 20 years ago, I was sitting in a physics class in high school when a "dumb" question occurred to me:
If two pendulums with unknown (different) frequencies started oscillating at the same time ...
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