# 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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### What could be the possible multiple of $x$ in the equation $12x = 5y^2$

If $x$ and $y$ are integers and $12x = 5y^2$, What could be the possible multiples of $x$ ? What can we infer about $y$ ? I have approached this problem like this : \begin{align} x = \dfrac{5y^2}{12}...
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### Largest possible prime factor for given $k$?

Let $k$ be a positive integer. What is the largest possible prime factor of a squarefree positive integer $\ n\$ with $\ \omega(n)=k\$ (That is, it has exactly $\ k\$ prime factors) satisfying the ...
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### How to understand special prime factorization method

Normally when we want to find the Prime Factorization of a number, we will keep dividing that number by the smallest prime number (2), until it can't be divided then we move on to the next prime ...
1 vote
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### Numbers for Testing Integer Factoring Algorithms

I'm looking for a list of numbers with which to test an integer factorization algorithm (for a computer). Something that has numbers harder than the ones I could easily come up with. Do any resources ...
1 vote
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### contest problem: number theory, prime factorization, perfect squares

Problem Statement: Gretchen labels each of the six faces of a cube with a distinct positive integer so that for each vertex of the cube, the product of the three numbers on the faces touching the ...
1 vote
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### Proof of uniqueness of prime factorization by induction with Euclid's Lemma [duplicate]

The uniqueness of the prime factorization is proved here on Wikipedia. I am wondering whether the initial trick could be extended to a full proof without using contradiction as it's done there. So, ...
1 vote
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### Can I make use of the "high" exponents here?

I currently try to factor the composite number $$3^{130}+5^{130}+7^{130}$$ and wonder whether the quadratic sieve can be accelerated because of the "high" exponents. I know that for numbers ...
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### Use Fermat-Kraitchik’s factorization to factor $85026567$

I have to use Fermat-Kraitchik’s factorization to factor $n=85026567$. I already know that $85026567$ is divisible by $3$, and therefore $85026567=3\cdot 28342189$, and both of the factors are prime. ...
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### Finding the prime factors of $2^{14}+3^{14}$ by hand

I need to find the prime factors of $2^{14}+3^{14}$ by hand (this was given in an exam at my university, so this is the motivation - I decided to state this because it may look unjustified to try to ...
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### An example showing $\mathbb{Z}[\sqrt[3]{7}]$ is not a UFD [closed]

It cannot be a UFD because it's the ring of integers of $\mathbb{Q}(\sqrt[3]{7})$ and has class number 3. How can we give an example showing this?
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If I partition an integer and get the prime factorization of each partition, is there a way to tell if my original integer was a prime? For example, given the factorization of my partitions $$71 = (56)... -1 votes 1 answer 99 views ### Properties of F(p)=p^2+1, where p is a prime number Let F(p)=p^2+1, where p is a prime number. For what primes p_1, p_2 does p_2 divide F(p_1) and p_1 divide F(p_2)? Two examples are \{p_1,p_2\}=\{5,13\}, and \{p_1,p_2\} = \{89,233\}... 2 votes 2 answers 53 views ### Prove that if k\mid q_1...q_n then we can find k_i such that k=k_1...k_n and for every i, k_i\mid q_i Let k=8 and q_1=12, q_2=2, q_3=5. We note that k\mid q_1q_2q_3, but k\nmid q_1 and k\nmid q_2 and k\nmid q_3. Also$$ 8=k_1k_2k_3=4*2*1,  where $k_1\mid q_1$, $k_2\mid q_2$, and $k_3\... • 597 0 votes 0 answers 32 views ### Factorization of Polynomials over$\Bbb Z_p\$ using GAP

I want to factorize Polynomials using GAP. Here is an example: ...