Questions tagged [prime-factorization]

For questions about factoring elements of rings into primes, or about the specific case of factoring natural numbers into primes.

A natural number is prime if it has no positive divisors besides $1$ and itself. The fundamental theorem of arithmetic states that every natural number $n>1$ can be factored uniquely, up to a reordering of the factors, as a product of distinct prime numbers each raised to some power.

This concept holds in a more general setting though. A ring is called a unique factorization domain (UFD) if every non-unit element can be factored uniquely as a product of prime elements in the ring. The ring of integers is an example of a UFD.

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If $\gcd(a, b) = p$, what are possible values of $\gcd(a^3, b^2)$?

If $\gcd(a, b) = p$, where $p$ is a prime. Then what are possible values of $\gcd(a^3, b^2)$? I have already calculated that $\gcd(a^2, b^2) = p^2$. $\gcd(a^2, b) = p$ or $p^2$, $\gcd(a^3, b) = p$ or $p^2$ or $p^3$. Can anybody give me any idea?…
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Prime elements of $\mathbb{Z}[i\sqrt5]$.

I was studying the Gaussian integers and I proved that every composite number in $\mathbb{N}$ is not a prime in $\mathbb{Z}[i]$. This is true because this ring is an Euclidean domain, and if $n=ab$ is composite…
user723846
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Question: If $p= n^{2} + 2 $ is a prime number, prove that n is a multiple of 3.

Helo guys! I have doubts on this question: "If $p= n^{2} + 2 $ is a prime number, prove that n is a multiple of 3." I did the test with n=3k (multiple of 3), n=3k+1 and n=3k+2, ($k \in R $)and I tried to show that in the last two cases $p$ was not…
Thais
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Find the greatest number that will divide 43, 91 and 183 so as to leave the same remainder in each case.

I KNOW THIS IS SOLUTION BUT I DON'T KNOW WHY? We first find the difference of the numbers and then find the HCF of the got numbers. 183−91=92 183−43=140 91−43=48 Now find HCF of 92, 140 and 48, we get 92=2×2×23 140=2×2×5×7 48=2×2×2×2×3 HCF(92, 140,…
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Solving the Diophantine equation $x^2 +2ax - y^2 = b$ in relation to Integer Factoring

Is there a method we could use for solving the Diophantine equation $x^2 +2ax - y^2 = b$? Background: Consider a very large integer $z$ that we want to factor (not necessarily prime factorization, but finding $pq = z$). Say we have $z$ unit square…
vvg
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Does knowing how $5280$ is a product of primes help you to factor $5281$ as a product of primes?

Does knowing how $5280$ is a product of primes help you to factor $5281$ as a product of primes? The above question was mentioned in Abstract Algebra book by Gallian. Prime factorization of $5280 = 2^5 .3.5.11$ $5281 = 5281$
Rkb
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what is a good estimation of $\rho(x)$ which is related to $\pi(x)$

what is a good estimation of $\rho(x)$. $\rho(x)$ is related to $\pi(x)$ the prime counting function? $\omega(x)$ is the number of prime factors of $x$ and $\sigma_0(x)$ is the number of factors of $x$ $$\rho(x)=\sum_{n=2}^{x}…
user813374
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Is $x^n - y^m$ irreducible ($n, m$ strictly positive integers with $n \ne m$)?

For $n$ and $m$ strictly positive integers with $n \ne m$, I guess that $x^m - y^n$ is irreducible in any field, but I am not sure. In particular, I have no idea how this could be proved. My question is thus: Is $x^m - y^m$ irreducible ? In any…
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Combinatorics and least common multiple

Would you be so kind as to provide me with a hint for a question that I can't solve? It is supposed to be more or less easy, but I don't see what the quick way to settle is. Let me thank you in advance for your help. Question. How many $4$-tuples…
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large factored number addition

I have some large integers that I want to add. The problem is, I can't even store them in a usual way. I store their prime decompositions, since they are so large. Here is one example of a product I…
Alex
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operations in determinant

I want to simplify a condition inside a determinant using coprime factorisations and I need to know if an operation is mathematically correct. This is a simplification of what I'm actually doing, but I only need to know if the last step is okay. I…
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About a basic property of prime numbers.

So I never really understood anything about prime numbers, other than the definition, and so I'm currently having problems with the following proposition: let $p$ be a prime number and $n,k\in\mathbb{N}$. If $k|p^n$, then $p|k$. I have no idea how…
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Disproving unique factorization of $\mathbb{Q}(\sqrt{-6})$

For homework, we were given a problem that asked to explain why $(\sqrt{-6})(-\sqrt{-6})$ implies $\mathbb{Q}(\sqrt{-6})$ does not have a unique factorization. I understand the unique part in this case only means unique up to multiplication by a…
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Why does $(5+\sqrt{3})(5-\sqrt{3})$ not conflict with $\mathbb{Q}(\sqrt{3})$ having unique factorization?

My intuition is to try to show that there is some irreducible element $p \in \mathbb{Q}(5)$ that divides $(5+\sqrt{3})(5-\sqrt{3})$, but I'm having trouble finding it. Is there an easier way to do this?
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how (a!)/(b!) = (b + 1)×(b + 2)×⋯×(a − 1)×a

I was solving a problem in which i need to figure out the prime factorization of $\frac{a!}{b!}$ and i did that by computing (a!) and then (b!) by looping ((1 to a) & (1 to b)) and then derived n by dividing them ($n = \frac{a!}{b!}$) and then prime…
gaurav
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