# Questions tagged [factorization]

For questions about factorization, the decomposition of mathematical objects (e.g. natural numbers, polynomials) into products of smaller objects (e.g. primes, lower degree polynomials).

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### Density of numbers with multiple factors near square root

Fix constants $1\leq \alpha<\beta$. What is the density of the set of positive integers $n$ with at least two factors between $\alpha\sqrt{n}$ and $\beta\sqrt{n}$?
(I am specifically interested ...

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### Probabilistic Howgrave-Graham Bounds with Coppersmith Technique?

Howgrave-Graham condition says roots of $$f(x_1,\dots,x_n)\equiv0\bmod R$$ at an $R\in\mathbb N$ are roots of $$f(x_1,\dots,x_n)=0$$ over $\mathbb Z$ if $$\|f(x_1X_1,\dots,x_nX_n)\|<\frac{R}{\sqrt{\...

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262 views

### Reason Coppersmith fails here?

Take classic problem of finding $P,Q$ in balanced semi-prime $N=PQ$.
$P$ has a binary expansion and so does $Q$. We can set the binary $0/1$ variables to be $x_1$ through $x_{\lceil\log P\rceil}$ and ...

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### Factoring with partial information on gaps

If $N=PQ$ is a semi-prime with $P=N^{\frac12 +\delta}$ and $Q=N^{\frac12-\delta}$ then if we know $\delta\in(0,\frac12)$ to a reasonable precision we can factor $N$ quickly. What precision (number of ...

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### Does Coppersmith technique suffice to factor?

Take classic problem of finding $P,Q$ in balanced semi-prime $N=PQ$. Is there evidence that no extension of Coppersmith technique will accomplish factoring $N=PQ$ in polynomial time?
Technically I am ...

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### Generalizing cycle/pseudo-tree factorizations for permutations/transformations to arbitrary binary relations

It's well known every permutation has a unique factorization into disjoint cycles (up to a re-ordering of these factors since they commute), while similarly it can be shown that every transformation ...

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### Coppersmith's method to quadrivariate degree $2$ polynomials that behave as bivariate?

We have a polynomial $f(x_1,x_2,x_3,x_4)\in\mathbb Z[x_1,x_2,x_3,x_4]$ where the only monomials are either from set $$\{x_1,x_1x_2,x_2,x_3,x_3x_4,x_4\}$$ and we seek solutions $(x_1,x_2,x_3,x_4)\in\...

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### Is total degree version and $x,y$ degree version of Coppersmith's theorem correct?

The notes here https://web.eecs.umich.edu/~cpeikert/lic13/lec04.pdf have the note 'Small decryption exponent $d$: so far the best known attack recovers $d$ if it is less than $N^{.292}$. This uses a ...

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### $\{ P_3, P_4 \}$-factor

Definition. A graph $G=(V,E)$ is to be $\{d_1,\dots,d_n\}$-graph if for each vertex $v\in V$ we have $\text{deg}(v)=d_i$ for some $i=1,\dots n$.
Definition. A connected graph $G=(V,E)$ is called $...

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### The power of a prime in the prime factorization of a factorial [closed]

How do we find—for example—how many $5$s are in the prime factorization of $n!$? I've read that it is $\lfloor n/5 \rfloor$, but why is that?

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### Finding a particular matrix factor

Consider the following Laurent polynomial matrix-valued function in the variable $x\in\mathbb{C}$
$$
A(x) = \begin{bmatrix} 0 & x \\ x^{-1} & 0\end{bmatrix}.
$$
I'm interested in finding a ...

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### Can we efficiently factor $n$ given that $n=pq$ where $p,q$ are primes satisfying $p=a^2+b^2, q=2ab+1$ for some $a,b$

Suppose we're given a particular number $n \in \mathbb{N}$.
We're also given that $n=pq$ where $p,q$ are unknown primes satisfying
$$
p=a^2+b^2
$$ and
$$
q=2ab+1
$$
for some $a,b$.
Is there an ...

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131 views

### Factoring certain Hessians of real homogeneous bivariate polynomials

For any homogeneous polynomial $f \in \mathbb R [x,y]$, define the homogeneous polynomial
$$H(f) := \partial_yf^2\partial_x\partial_xf-2\partial_xf\partial_yf\;\partial_x\partial_yf+\partial_xf^2\...

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### Factoring a positive semidefinite matrix into binary matrices

This question is motivated by a research problem I recently encountered. Consider two sets of random variables $\mathbf{X}$ and $\mathbf{Y}$, where $\mathbf{Y}$ can be expressed as a linear ...

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### List of analytically known eigensystems?

In condensed matter physics, we often come across matrices that are multi-diagonal or banded. For example, I may have a matrix with three tridiagonal bands, or a tridiagonal band and two/four ...

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### What is the natural density of hyper prime numbers?

What do we mean by hyper prime numbers? Well, roughly speaking they are natural numbers which are prime with respect to hyperoperators in arithmetic such as exponentiation, tetration, pentation, et ...

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### On the polynomial $\sum_{k=0}^n\binom{n}{k}(-1)^kX^{k(n-k)}$

Let $n = 2m$ be an even integer and let $F_n(X)$ be the polynomial $$F_n(X):=\sum_{k=0}^n\binom{n}{k}(-1)^kX^{k(n-k)}.$$ I observed (but cannot prove) that the polynomial $F_n$ is always divisible by $...

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### Factorization of polynomials into “shortest possible” factors

A while ago I asked a question at Mathematica.SE about how to factorize a polynomial into terms with as few monomials as possible each. I now realized that I actually do not know what is rigorous ...

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### A Handbook of Matrix Factorizations

I am looking for a good collection of facts regarding the various types of matrix factorizations, something like a "Handbook of Matrix Factorizations" or a very-thorough review paper. I am hoping for ...

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### amalgamated sum of monoids

Consider the amalgamated sum $Q_1 \rightarrow^{v_1} Q_1 \oplus_P Q_2 \leftarrow^{v_2} Q_2$ of $Q_1 \leftarrow^{u_1} P \rightarrow^{u_2} Q_2$ with $Q_1,Q_2,P$ being monoids.
Why does $v:= v_i \circ u_i$...

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### What size factor-base for Lenstra Elliptic Curve factorization

I'm writing a program to factorize numbers using Lenstra Elliptic Curve Factorization.
According to the wikipedia article, I should pick some k with a lot of small factors and then take a random ...

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169 views

### Factorization of Gabriel-Zisman localization construction?

My question concerns whether the Gabriel-Zisman localization construction $S^{-1}$ for categories admits a known factorization into a pair of commuting constructions $S^l$ and $S^r$.
The localization ...

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### Can a squarefree polynomial in K[x,y] not be squarefree in K[[x]][y]?

In a UFD, as usual one says that $f$ is square-free if it is not divisible by the square of any irreducible element, i.e., if it has no multiple factor.
An polynomial $f\in k[x,y]$ can have more ...

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### Conjecturally unsafe RSA primes $p=27a^2+27a+7$

We got strong numerical evidence that primes of the form $p=27a^2+27a+7$
are unsafe for cryptographic purposes since they can be found in the factorization.
Consider the following generic factoring ...

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227 views

### Which composites pass this probabilistic primality test?

If a composite integer resembles a prime too closely, it must pass
algorithmic tests designed to find primes and in addition avoid nontrivial
factorization.
Given an integer $p$, assume it is prime ...

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227 views

### How hard is it to compute these prime factor related problems?

We know that computing number of prime factors implies efficient factoring algorithm (How hard is it to compute the number of prime factors of a given integer?).
Let $\omega(n)$ be number of distinct ...

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### Equal degree factoring of homogeneous polynomials over $\Bbb Q[x_1,\dots,x_n]$?

Given $f(x_1,\dots,x_n)\in\Bbb Q[x_1,\dots,x_n]$ of form $\prod_{i=1}^df_i(x_1,\dots,x_n)$ where each of $f,f_i$ are homogeneous and each $f_i$ are irreducible and of equal degree what is the best ...

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### What relation does this problem have to Number Field Sieve?

In here Small geometric progression modulo N it is asked 'Must there exist a 5-term geometric progression $\lbrace a_0,a_1,a_2,a_3,a_4\rbrace$ (mod $N$) such that each term is $O(N^{2/3})$?
We also ...

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### Finding Rational Curves on a Surface

Let the field of rational numbers be our base field $k$. I hope to find all rational curves on the following surface $S$ defined by $f$. You can find the motivation in the end.
$f= (x^2y^2)z^3 + (5x^...

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### Finding numbers in specified interval with specified number of divisors

Problem: given an interval $[M_0,M_1]\subset Z_+$ and $D\in Z_+$, generate list of $x\in [M_0,M_1]$ such that $\tau(x)=D$. (here, $\tau$ is the number-of-divisors function).
Question 1: is there any ...

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683 views

### Hilbert Numbers

A positive integer $n$ is called a Hilbert number if $\exists a,b,d \in \mathbb{N}$ such that $ 4ab-a-b = d n$ and $d|a b$.
I ran an algorithm checking divisors for all $0\lt a,b\le500$, and the ...

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### how two orthonormal function multiplication can be represented as a function of their arguments difference

Can a pair of orthonormal signals ${{\phi }_{k}}\left( t \right),\phi _{n}^{{}}\left( t \right)$ admit the following relation
${{\phi }_{k}}\left( t \right)\phi _{n}^{*}\left( s \right)+{{\phi }_{...

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### $\log \log p / \log \log n$, where $p|n$, gets equidistributed in [0,1] (for almost all $n$)

According to Hardy-Ramanujan/Erdős-Kac we know that usually there are $\sim\log\log n$ prime numbers in a factorization. But if you pick up a natural number at random, and you factor it, what is the ...

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### Irreducible Hurwitz Factorization of A Complex Polynomial

I've decided to repost this question, which originally appeared on MSE, here. It is part of my series of open problems for enthusiasts and, while I understand this crowd is focused on professionals, ...

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### Factorization of trigonometric matrices

Consider two real square matrices $A_1$ and $A_2$ and $t_1,t_2\in\mathbb{R}$. $A_1$ and $A_2$ do not commute. Consider the following matrix involving matrix trigonometric functions:
\begin{equation}
...

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### Check irreducibility of an explicit polynomial, without computer

I have a polynomial of degree 8 in 6 variables given explicitly by
$$ (\sqrt{1+(x_1+x_2+x_3)^2+(y_1+y_2+y_3)^2}+\sqrt{1+x_1^2+y_1^2}+\sqrt{1+x_2^2+y_2^2}+\sqrt{1+x_3^2+y_3^2})\times\text{the other ...

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### Current best time for factoring in $\Bbb Q[x]$

Lenstra Lenstra Lovasz have a $O((nb)^{11})$ deterministic algorithm to factor primitive polynomials in $\Bbb Q[x]$ where $b$ is total number of bits in the polynomial and $n$ is degree of the ...

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### Generalizing Kasteleyn's formula even more?

Inspired and intrigued by this question, I decided just for fun to throw in another integer into the factors and look what happens. So for $k\in\mathbb Z$, let us define $$K_r(n,k):=\prod_{\ell_1=1}^...

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### Counting integers with k large prime divisors

If $x \ge y \ge 1$ are real numbers and if $k$ is a positive integer, take $\Phi_k(x, y)$ to be the number of integers $\le x$ with exactly $k$ prime factors and no prime factor $\le y$. If $y$ is ...

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### Upper bound for tuple of exponents of prime factorization

Let $a(n)$ be the $k$-ary tuple of the exponents of the prime factorization of $n$. For example,
$$a(5184)=a(2^{6}⋅3^{4})=(6, 4), a(65536)=a(2^{16})=(16).$$
Formally, let $p_{1}^{a_{1}}, p_{2}^{a_{2}...

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### Upper bound for product of exponents of prime factorization

Let $p(n)$ be the product of the exponents of the prime factorization of $n$. For example,
$$p(5184)=p(2^{6}\cdot 3^{4})=24,\qquad
p(65536)=p(2^{16})=16.$$
Is $p(n) = O(\log^{k}(n))$ for some constant ...

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192 views

### Is it possible to show that :for $n \geq 1:\sigma(n!-1) $ never be prime and why $\sigma(n!-1)\bmod 10 $ at most is $0$?

This question is related to my question here , I w'd like to check if $n \geq 1:\sigma(n!-1) $ never be prime according to some computations which i did in wolfram alpha to come up with parity of sum ...

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### How much space between these smooth numbers?

In looking at OEIS sequence A063539, $1,8,12,16,18,24,27,30,32,36,40,45,...$ I noticed that the first 1000 members were less than 4000, and thought there were no large gaps between them. What (if ...

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### Structures in the plot of the “squareness” of numbers

(This is based on an earlier MSE posting,
"Structures in the plot of the “squareness” of numbers.")
My main question is to explain the structural features of this plot:
This is a plot of what I call ...

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### Factorially closed subrings

Lemma 3.2 says: Let $A$ be a UFD. Let $R \subseteq A$ be a subring of $A$ such that
$R^* = A^*$.
The following conditions are equivalent:
(i) Every irreducible element of $R$ remains irreducible in $...

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### How can one construct a sparse null space basis using recursive LU decomposition?

Given an $m$ by $n$ matrix $A$ I'm familiar with the standard method to compute a basis for the null space of $A$ by computing a QR factorization of $A^T$. If $A$ is large and sparse, we can use ...

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### Unique factorization for the semigroup generated by {2cos(π/n) | n>3}?

Let $S$ be the multiplicative semigroup of numbers generated by $B=\{ 2cos(\frac{\pi}{n}) \mid n \ge 4 \}$.
Question: Does every number of $S$ factorize uniquely (up to perm.) as a product of ...

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237 views

### Is there a “small $\omega$” number theorem?

In my studies of how primes jump (search this forum for a link), a question has been raised which may have been studied. Can anyone jump-start my literature search with references regarding the ...

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230 views

### Is a category regular iff extremal epis and monomorphisms are a strong factorization system?

I'm reviewing some notes of mine on regular categories to try and get a better feel for regular, strong, and extremal epimorphisms, and this leads me to ask: is a category $\mathsf C$ regular if and ...

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622 views

### Results on the largest prime factor of $2^n+1$

A work of Cameron Stewart (the paper has appeared in Acta Mathematica), proving a conjecture of Erdos, Stewart shows that
the largest prime factor of $2^n-1$ is at least
$n \times \exp\Big( \frac{\...