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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Minimum $k$ for which every positive integer of the interval $(kn,(k+1)n)$ is composite
I am looking for references containing results on the minimum $k$ for which every positive integer of the interval $(kn,(k+1)n)$ is composite.
If we denote as $k(n)$ this minimum $k$ for some $n$, $k$ ...
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Existence of prime elements in an atomic integral domain
Let $R$ be an integral domain, is it true that if $R$ is atomic, then it must contain a prime element?
If not, what is a counterexample?
I know that if an element is prime, then if $I$ is the ideal ...
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Denjoy's Probabilistic Interpretation
Does Denjoy's Probabilistic Interpretation actually "prove" that the Mertens function ratio between numbers with odd number of distinct prime factors and even number of prime factors is 1? ...
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Prime Divisor of the Sum of Two Squares
I'm struggling something immensely to make sense of the following:
https://meiji163.github.io/post/sum-of-squares/#sums-of-two-squares
Factoring an integer in Gaussian integers is closely related to ...
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The chunking aspect of repunit prime factors [closed]
While others have already mentioned the divisibility of decimal repunits,
$$m := \frac{{10}^n - 1}{9}.$$
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Will there be infinity prime numbers of the sort $a^2 -2$ (where $a$ is odd)?
Will there be infinity prime numbers of the sort $a^2 -2$ (where $a$ is odd)?
To begin with, every odd composite number can be written as $a^2$ or as $a_{x}^2 -a_{y}^2$ as long as either $x$ or $y$ ...
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Problem in understanding the unique factorization theorem for Euclidean Rings.
Unique Factorisation Theorem: Let $R$ be a Euclidean ring and $a\neq 0$ non-unit in $R.$ Suppose that $a =\pi_1\pi_2\cdots\pi_n=\pi_1'\pi_2'\cdots\pi_m'.$ where the $\pi_i$ and $\pi_j'$ are prime ...
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Unique unramified ideal implies that the ramification index is equal to the degree of field extension in a galois extension
Given a Galois extension $K \supseteq \mathbb{Q} $, prove that if there is only one unramified prime number $p$ over $K$ then there is only one prime ideal $\mathfrak{p} \subseteq O_K$ containing $p$ ...
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Can you prove that 2^n-1 will be divisible by 3 if n is even. [duplicate]
Can you prove that 2^n-1 will be divisible by 3 if n is even.
I have generated this:
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Prime number $p$ such that $p+1$ has all given prime numbers as prime factor.
For given finite prime numbers set $P$,
does there exist some prime number $p$ such that
for any $\ell\in P$, $\ell\mid (p+1)$?
For example, if $P=\{2,3,7\}$, then we can take $p=41$.
In this case, $(\...
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For what integers $n$ does $\varphi(n)=n-5$?
What I have tried so far: $n$ certainly can't be prime. It also can't be a power of prime as $\varphi(p^k)=p^k-p^{k-1})$ unless it is $25=5^2$. From here on, I am pretty stuck. I tried considering the ...
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What did I get wrong in this Mobius function question? [closed]
$f(n):=\sum\limits_{d\mid n}\mu(d)\cdot d^2,$ where $\mu(n)$ is the Möbius function. Compute $f(192).$
First, I found all of the divisors of 192 by trial division by primes in ascending order:
$D=\{...
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What are the next primes/semiprimes of the form $\frac{(n-1)^n+1}{n^2}$?
This question is inspired by this question
For an odd positive integer $n$ , define $$f(n):=\frac{(n-1)^n+1}{n^2}$$ as in the linkes question.
For which $n$ is this expression prime , for which $n$ ...
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Factorization of Proth numbers with $k=1$
This is more of a practical question, for anyone out there who might know where to start. I'm looking for a complete factorization of numbers of the form $2^n+1$ for positive integers $n$. Essentially ...
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Splits completely of a prime ideal
Suppose that $K$ is a number field and $\mathfrak p$ is a prime ideal non-zero.
In general always exists a finite extension of $L$ of $K$ such that $\mathfrak p$ is ramified, for example $L=K(\sqrt f)$...
