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).
5
votes
1answer
85 views
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 ...
0
votes
0answers
72 views
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 $$\|f(x_1X_1,\dots,x_nX_n)\|<\frac{R}{\sqrt{\...
0
votes
1answer
262 views
Reason Coppersmith fails here?
Take classic problem of finding $P,Q$ in balanced semi-prime $N=PQ$.
$P$ has a binary expansion and so does $Q$. We can set the binary $0/1$ variables to be $x_1$ through $x_{\lceil\log P\rceil}$ and ...
1
vote
1answer
97 views
Factoring with partial information on gaps
If $N=PQ$ is a semi-prime with $P=N^{\frac12 +\delta}$ and $Q=N^{\frac12-\delta}$ then if we know $\delta\in(0,\frac12)$ to a reasonable precision we can factor $N$ quickly. What precision (number of ...
0
votes
0answers
104 views
Does Coppersmith technique suffice to factor?
Take classic problem of finding $P,Q$ in balanced semi-prime $N=PQ$. Is there evidence that no extension of Coppersmith technique will accomplish factoring $N=PQ$ in polynomial time?
Technically I am ...
1
vote
1answer
54 views
Generalizing cycle/pseudo-tree factorizations for permutations/transformations to arbitrary binary relations
It's well known every permutation has a unique factorization into disjoint cycles (up to a re-ordering of these factors since they commute), while similarly it can be shown that every transformation ...
2
votes
0answers
86 views
Coppersmith's method to quadrivariate degree $2$ polynomials that behave as bivariate?
We have a polynomial $f(x_1,x_2,x_3,x_4)\in\mathbb Z[x_1,x_2,x_3,x_4]$ where the only monomials are either from set $$\{x_1,x_1x_2,x_2,x_3,x_3x_4,x_4\}$$ and we seek solutions $(x_1,x_2,x_3,x_4)\in\...
4
votes
1answer
219 views
Is total degree version and $x,y$ degree version of Coppersmith's theorem correct?
The notes here https://web.eecs.umich.edu/~cpeikert/lic13/lec04.pdf have the note 'Small decryption exponent $d$: so far the best known attack recovers $d$ if it is less than $N^{.292}$. This uses a ...
5
votes
1answer
160 views
$\{ P_3, P_4 \}$-factor
Definition. A graph $G=(V,E)$ is to be $\{d_1,\dots,d_n\}$-graph if for each vertex $v\in V$ we have $\text{deg}(v)=d_i$ for some $i=1,\dots n$.
Definition. A connected graph $G=(V,E)$ is called $...
-1
votes
1answer
132 views
The power of a prime in the prime factorization of a factorial [closed]
How do we find—for example—how many $5$s are in the prime factorization of $n!$? I've read that it is $\lfloor n/5 \rfloor$, but why is that?
5
votes
1answer
130 views
Finding a particular matrix factor
Consider the following Laurent polynomial matrix-valued function in the variable $x\in\mathbb{C}$
$$
A(x) = \begin{bmatrix} 0 & x \\ x^{-1} & 0\end{bmatrix}.
$$
I'm interested in finding a ...
2
votes
0answers
193 views
Can we efficiently factor $n$ given that $n=pq$ where $p,q$ are primes satisfying $p=a^2+b^2, q=2ab+1$ for some $a,b$
Suppose we're given a particular number $n \in \mathbb{N}$.
We're also given that $n=pq$ where $p,q$ are unknown primes satisfying
$$
p=a^2+b^2
$$ and
$$
q=2ab+1
$$
for some $a,b$.
Is there an ...
3
votes
2answers
131 views
Factoring certain Hessians of real homogeneous bivariate polynomials
For any homogeneous polynomial $f \in \mathbb R [x,y]$, define the homogeneous polynomial
$$H(f) := \partial_yf^2\partial_x\partial_xf-2\partial_xf\partial_yf\;\partial_x\partial_yf+\partial_xf^2\...
5
votes
2answers
188 views
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 ...
2
votes
0answers
128 views
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 ...
5
votes
1answer
425 views
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 ...
33
votes
7answers
1k views
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 $...
12
votes
1answer
262 views
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 ...
4
votes
1answer
157 views
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 ...
1
vote
1answer
69 views
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 $v:= v_i \circ u_i$...
1
vote
0answers
64 views
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 ...
5
votes
1answer
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 ...
8
votes
1answer
395 views
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 ...
44
votes
1answer
16k views
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 ...
2
votes
1answer
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 ...
2
votes
1answer
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 ...
1
vote
1answer
169 views
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 ...
0
votes
0answers
115 views
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 ...
2
votes
0answers
138 views
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^...
0
votes
0answers
69 views
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 ...
4
votes
2answers
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 ...
0
votes
0answers
47 views
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 }_{...
7
votes
1answer
296 views
$\log \log p / \log \log n$, 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 ...
4
votes
1answer
133 views
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, ...
2
votes
1answer
140 views
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}
...
6
votes
1answer
245 views
Check irreducibility of an explicit polynomial, without computer
I have a polynomial of degree 8 in 6 variables given explicitly by
$$ (\sqrt{1+(x_1+x_2+x_3)^2+(y_1+y_2+y_3)^2}+\sqrt{1+x_1^2+y_1^2}+\sqrt{1+x_2^2+y_2^2}+\sqrt{1+x_3^2+y_3^2})\times\text{the other ...
3
votes
0answers
82 views
Current best time for factoring in $\Bbb Q[x]$
Lenstra Lenstra Lovasz have a $O((nb)^{11})$ deterministic algorithm to factor primitive polynomials in $\Bbb Q[x]$ where $b$ is total number of bits in the polynomial and $n$ is degree of the ...
4
votes
1answer
222 views
Generalizing Kasteleyn's formula even more?
Inspired and intrigued by this question, I decided just for fun to throw in another integer into the factors and look what happens. So for $k\in\mathbb Z$, let us define $$K_r(n,k):=\prod_{\ell_1=1}^...
4
votes
1answer
123 views
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 ...
0
votes
1answer
90 views
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}...
2
votes
1answer
205 views
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 ...
-2
votes
3answers
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 ...
1
vote
1answer
263 views
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 ...
26
votes
4answers
2k views
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 ...
2
votes
0answers
85 views
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 $...
7
votes
2answers
923 views
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 ...
4
votes
0answers
150 views
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 ...
2
votes
1answer
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 ...
3
votes
1answer
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 ...
7
votes
1answer
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{\...
