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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$x^2+y^2=p$ has a solution in $\mathbb{Z}$
Show that $x^2+y^2=p$ has a solution in $\mathbb{Z}$ if and only if $ p≡1 \mod 4$. Thnx, if someone can help
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$K[x_1, x_2,\dots ]$ is a UFD
I wonder about how to conclude that $R=K[x_1, x_2,\dots ]$ is a UFD for $K$ a field.
If $f\in R$ then $f$ is a polynomial in only finitely many variables, how do I prove that any factorization of $f$...
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Redistribution/synchronization problem
I have a very peculiar problem.
I am interested to solve problems such as:
B>8 (first occurence of B>8) for 40*B = 15*D such that B and D are integer numbers.
The problem is also illustrated in the ...
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I found out that $p^n$ only has the factors ${p^{n-1}, p^{n-2}, \ldots p^0=1}$, is there a reason why?
So I've known this for a while, and only finally thought to ask about it.. so, any prime number ($p$) to a power $n$ has the factors $\{p^{n-1},\ p^{n-2},\ ...\ p^1,\ p^0 = 1\}$
So, e.g., $5^4 = 625$,...
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$\log$ transform of the fundamental theorem of arithmetic? [closed]
Taking the canonical form of the fundamental theorem of arithmetic in the form:
$$n=\prod_{j=1}^\infty p^{m_j}_j \qquad ;m_j\in \Bbb N_0$$
Does anybody know about a $\log n$ transform of this?
Note: ...
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Number of distinct prime factors, omega(n)
Is there a formula that can tell us how many distinct prime factors a number has?
We have closed form solutions for the number of factors a number has and the sum of those factors but not the number ...
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Combination of positive integers in terms of primes (sophisticated version 2)
Here comes a second sophisticated version of my conjecture, as critics came up the first version was trivial.
Teorem 2
for a given prime $p$ and a given power $m$ the representation of any positive ...
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Conjecture on combinate of positive integers in terms of primes
Along a heuristic calculation, I am struggeling with a possible proof for my following conjecture:
Every positive integer $n\in \Bbb N$ can be written as a unique combination of $a,b \in \Bbb N$, $m\...
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Divisibility and factors [duplicate]
1) Can factors be negative? Please prove your opinion.
2)If prime factorization is given to you, how will you find out how many composite factors are there? Not the factors, just how many.
For 2), my ...
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If every irreducible element in $D$ is prime, then $D$ has the unique factorization property.
Suppose every irreducible element in a domain $D$ is prime.
I'm trying to prove this implication:
In a integral domain $D$, if $a=c_1c_2...c_n$ and $a=d_1d_2...d_m$
($c_i,d_i$ irreducible), then ...
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How do we find the prime ideals of a ring of integers of a number field?
How do we find the prime ideals of a ring of integers of a number field?
For example, for $F=\mathbb Q(\sqrt{-5})$ the ring of integers of $F$ is $\mathbb Z[\sqrt{-5}]$ (since $-5\equiv3 \pmod 4$). ...
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Negative factors of a number
Can a factor of a number be negative?
Is $-5$ a factor of $25$ or $-25$?
A number is said to be prime if it has two factors : $1$ and the number itself. So if $-5$ can be a factor of $5$, how to ...
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Prime divisibility in a prime square bandtwidth
I am seeking your support for proving (or fail) formally the following homework:
Let $p_j\in\Bbb P$ a prime, then any $q\in\Bbb N$ within the interval $p_j<q<p_j^2$ is prime, if and only if all:...
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Classification of nonzero prime ideals of $\mathbb{Z}[i]$
I know the classification of Gaussian primes: let $u$ be a unit of $\mathbb{Z}[i]$. Then the following are all Gaussian primes:
1) $u(1+i)$
2) $u(a+ib)$ where $a^2+b^2=p$ for some prime number p ...
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How can can you write a prime number as a product of prime numbers?
According to the fundamental theorem of arithmetic (unique factorization theorem), you can write every number as the product of some prime numbers, for example $33 = 11 \cdot 3$.
However, how can you ...
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