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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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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Is there anything I could read that talks about dimensionality of prime/composite numbers?
Is there anything out there that talks about how primes are one dimensional numbers and composites can only be in dimensions greater than 1?
What I mean is, 4 would be a two dimensional number (2x2) ...
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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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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 $L=K(\sqrt f)$...
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$3^p-2^p$ squarefree?
Original question:
$3^n-2^{n-1}$ seems to be squarefree. Is it?
Answer: No, but amongst the primes dividing one of these numbers, $23$ seems to be a special case: no $23^2$ divide any of them
Are ...
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Finding positive integer $n>10$ that maximizes $\frac{\sigma_0(n)}{2^{\log n}}$
Among all the positive integer, which one integer, $n$, can make the number below the largest?
$$f(n)=\frac{\sigma_0(n)}{2^t}$$where $t=\log_{10}n$ and $\sigma_0$ is the divisor function.
For example,...
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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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Minimising $x+y$ in $x^y=a$
Regarding this recent question, the question asks for minimizing $x+y$, given $x^y=a$, (for a constant real positive $a$) for real and positive values of $x$ and $y$.
Now $(x,y)=\left(\exp\left({2W\...
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Plot of the ratios of Goldbach pairs
Preface
I was playing around with matplotlib to generate some number sequences. I wound up looking at Goldbach pairs and manipulating them in different ways. End result was the following plots. I can'...
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