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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).

4
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1answer
192 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
157 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
126 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
128 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
186 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
125 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
185 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
126 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
419 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 ...
31
votes
6answers
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
260 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
150 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
67 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
62 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
163 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
385 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
224 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 ...
0
votes
0answers
128 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
165 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
137 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
678 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
291 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
130 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
139 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
244 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
81 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
121 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
200 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
189 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
262 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
82 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
853 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
149 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
232 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
221 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
608 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{\...
3
votes
1answer
190 views

Irreducibility of family of polynomials

Consider the following family of polynomials over $\mathbb{Q}$: $$f_n = x^n - x^{n-1} - \dots - 1$$ Notice that these polynomials satisfy the recurrence $$ f_{n+1} = x f_n - 1 $$ I would like to ...
11
votes
0answers
154 views

Are there efficient algorithms to factorise in $\mathbb{N}[X]$?

One way to do factorisation in $\mathbb{N}[X]$ is to use an algorithm to factorise in $\mathbb{Z}[X]$ and then to combine some factor to find a factorisation in $\mathbb{N}[X]$. Note that the ...
9
votes
1answer
302 views

Intuition for density comonad in relation to lifting problems

In Emily Riehl's Categorical Homotopy Theory, there is a section on Garner's Small Object Argument which I'm trying and failing to understand. Originally I followed most of Garner's paper, using the ...
0
votes
0answers
124 views

More generalized RSA construction

Is there a way to construct RSA type cryptosystem over general number rings? Can Number Field Sieve technique be applied here?
2
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2answers
84 views

Solution of $\left(\Gamma(x+c)/ \Gamma(x+d)\right)y^d/y^c = {\rm const}$

When I try to solve $F(x,y)= \Gamma(x+c)/\Gamma(x+d) y^d/y^c = {\rm const}$, I find that $y = p x + q$ satisfies the above equation, whith specific $p$ and $q$ constants for the given constants $c$ ...
3
votes
1answer
127 views

Reference Request on the existence of $k$ satisfying $P(\Phi_d(2))^k \gt \Phi_d(2)$ for all $d$

I am working my way through the literature regarding the following conjecture: There is a positive integer $k$ such that for all positive integers $d$, $$P(\Phi_d(2))^k \gt \Phi_d(2).$$ I am ...
1
vote
0answers
110 views

Expressing Numbers with a Minimal Sum in Powers of 2 [closed]

The first 64 bits of pi are: 11.00100100001111110110101010001000100001011010001100001000110100 Computer multiplication can be sped up by looking for patterns and ...