# 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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### Number of right divisors of a central skew polynomial

Let $\mathbb{F}$ be a finite field of $p$ elements, $\sigma \in \operatorname{Aut}(F)$ of order $m$, $\mathbb{F}^\sigma$ be the fixed field of $\sigma$, and $\mathbb{F}[x,\sigma]$ be a skew polynomial ...

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### Factor group of all the sequences by the subgroup of bounded sequences

Consider the group G of the sequences of real numbers (the group operation is addition). It contains a subgroup H of bounded sequences.
Is there any nice description of the factor group G/H ?
It is ...

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### $p^2+a^2$ can be a squarefree number with all prime divisors less than $p$?

Let $p$ be a prime $\ge 31$.
Is there an integer $a < p$ such that $p^2 + a^2$ is a squarefree and all of its prime divisors are less than $p$?
For example, for $p=31$, $31^2+5^2 = 986 = 2 \times ...

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### weak factorization systems (co)generated by an arbitrary class of morphisms

Under what assumptions can one prove existence of weak factorization systems (co)generated by a class of morphisms ?
Are there counterexamples ? I am interested both in assumptions on the class of ...

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### Optimal Truncation of LDL-factorization to improve conditioning

Suppose I factored real symmetric quasi-definite $ A_0= L_0 \cdot D_0 \cdot L_0^T$ and the factorization exists, with $D$ diagonal and $L$ unit lower-triangular; and suppose $L$ and $D$ are badly ...

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### Slope assertion in Cholesky on digital computers

For a real symmetric positive definite linear system
$$ A \cdot x = b, $$
solved using Choelsky with forward- and backward-substitution, we know it for the numerical approximation $\tilde{x}$ to $x$ ...

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### Can nonnegative functions $f(x,y,z)$ be written as a product of pairwise functions $u(x,y) v(y,z) w(x, z)$?

In my course on probabilistic graphical models, my professor made a claim which I find a little sus. In discussing the equivalence between Markov Random Fields and Factor Graphs, the following example ...

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### Practical symmetric equivalent to QR factorization updates

As we know, the QR-factorization $Q\cdot R=A$ of any real symmetric $n \times n$ matrix $A$ with full rank is unconditionally numerically stable. Further, when A is rank-1-updated, the factorization ...

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### Modular square roots problem which is $NP$ hard

It is well known extracting modular square roots modulo a composite number factors the modulus.
On other hand given $u,v>0$ and an integer $n$, deciding if there is a factor of $n$ in $[u,v]$ is $...

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### Twin prime distribution centering twice a semiprime

What is the conjectured distributional behavior of semiprimes $pq$ ($p$ and $q$ are primes) having the property $2pq+1$ and $2pq-1$ are primes?

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### Distribution of number of prime factors of $p^k\pm1$

What is the behavior of number of prime factors of integers of form $p^k\pm1$ where $p$ is a fixed odd prime or $2$ and $k$ varies over positive integers?

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### Is this factorization problem in EXP?

Factorization is not known to have a polynomial time algorithm. Traditionally the input length is number of bits in representation of the integer to be factored.
However now consider integers of form $...

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### How to decompose a matrix over a ring $F[X_1,\ldots,X_k]$ as a product of two matrices

Let $F$ be a field. Assume any reasonable conditions if needed, such as $F=\mathbb R$, $F=\mathbb C$, $F$ is a finite field, or $F$ has a specific characteristic, etc. Let $C$ be an $n\times1$ matrix ...

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### Could a quantum computer factor $N=p\times q$ using Hadamard transforms on $x^2\bmod N$ (instead of Fourier transforms on $a^x\bmod N$)?

In Classically verifiable quantum advantage from a computational Bell test, Kahanamoku-Meyer, Choi, Vazirani, and Yao propose using $x^2 \bmod N$ in an interactive proof-of-quantumness. This is a two-...

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### How many elements have a "small" order in a finite field?

I'm hoping that this is an easy question for someone.
How many elements can we expect to have multiplicative order at most $n^{1/c}$ in one of the finite fields $\mathbb{F}_p$ with $p$ prime with $n \...

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### Identifying redundant vectors In non-negative matrix bases

I have a target non-negative matrix $X$ that I would like to factor.
I have two non-negative matrices $W$ and $H$ such that $WH = X$. In this formulation, the rows of $H$ are $L^2$ normalized and ...

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### Atomicity and BF-ness in monoids of integer points of a polyhedral cone of $\mathbb R^n$

Fix an integer $n \ge 2$ and let $H$ be the (additive) monoid of integer points of a polyhedral cone of the Euclidean space $\mathbb R^n$ with the additional property that $H \setminus \{0_n\}$ is ...

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### What are the limitations for calculating the inverse of a polynomial with the Lagrange inversion theorem?

I have been attempting to produce a series expression for the roots of high degree polynomial using the Lagrange Inversion theorem. I am curious about the statement from the Wikipedia page on Bring ...

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### Characterizing atomicity in a commutative domain

In Proposition 1.1 of [Math. Proc. Cambridge Phil. Soc. 64 (1968), No. 2, 251-264], P.M. Cohn famously claimed (without proof) that a commutative domain is atomic if and only if it satisfies the ...

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### Removing intermediate terms from a $10^{\text{th}}$-degree polynomial

When given a tenth degree polynomial
$$f(x)=\sum_{n=0}^{10} a_n x^n$$
I wish to compute an inverse via the Lagrange inversion theorem to produce a generalized hypergeometric sum similar to a Bring ...

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### Polynomial divisible by unbounded primes with exponent one

Let $f(x)$ be squarefree polynomial with integer coefficients and
degree at least $3$.
Is it true that for all sufficiently large $n$, $f(n)$ is divisible
by prime $p$ with exponent one and $p$ is ...

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### On the sequence $a(n)=\gcd(2^n-1,\phi(2^n-1))$

For natural $n$, define the sequence
$$
a(n)=\gcd(2^n-1,\phi(2^n-1))
$$
It doesn't appear to be in OEIS and starts
$1,1,1,1,9,1,1,1,3,1,9,1,3,1,1,1,27,1,75,49$
Q1 Can we unconditionally prove $a(n)=1$...

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### Factor $\sum_{n=1}^{N} x^n$ [closed]

I am attempting to factor an $N^{\text{th}}$ degree polynomial with coefficients strictly equal to $1$ given by the equation
$$\sum_{n=1}^{N} x^n$$
Although the Galois group for anything beyond a ...

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### Do polynomial values rarely have large multiple prime factors?

I am interested in the following set-up:
Let $F \in \mathbb{Z}[x_1,\dots,x_n]$ be a fixed irreducible homogeneous polynomial of degree $d$ and consider the quantity
$$N_{\delta}(B)=\#\{(x_1,\dots,x_n) ...

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### A gsvd variation: Two SVDs with common matrix

The generalized singular value decomposition (gsvd) is described on wikipedia here. Since there are several conventions, I'll just briefly present one.
The "MATLAB" convention decomposes two ...

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### Construct special "joint SVD" from separate SVDs

Given two matrices, $A,B\in\mathbb{C}^{n\times n}$ which can be written as
$$ A = XD_AY^T \\
B = XD_BY^T $$
where $X$ and $Y\in\mathbb{C}^{n\times n}$ are unitary and with diagonal $D_A$ and $D_B\in\...

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### Order of elements $\gamma$ and $1-\gamma$ in $\mathbb{F}_q$

I'd like to find if possible the orders possible for $1-\gamma$ given a $\gamma\in\mathbb{F}_q$ of given order $\mathcal{o}$ (where $q=p^f$).
Which is clear is that these order have the same degree (...

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### Order of roots for a polynomial $P\in\mathbb{F}_p[T]$

Let $P\in\mathbb{F}_p[T]$ (not supposed irreducible). All roots $\xi$ of $P$ have a certain order $k$ such that $\xi^k=1$.
Question: is it possible to know the order of the roots of the given ...

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### From direct sum of quotient group of a group to direct sum of the group

We know that for a cyclic group $G$, if $G=A\oplus B$, then for some subgroups $H$ of $G$, We have $G/H=(A+H)/H\oplus (B+H)/H.$ But, if we know that for a subgroup $H$ of $G$, $G/H=(A+H)/H\oplus (B+H)...

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### What is the factorization algebra/space of an affine W algebra?

The affine vertex algebra $V_k(\mathfrak{g})$ factorizes, i.e. comes from a factorisation space, the Beilinson Drinfeld Grassmannian. Similarly, lattice vertex algebras have a factorization analogue.
...

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### Eisenstein polynomial of totally ramified extension over $p$-adic field

Let $p\geq 3$ be a prime number, $K$ be a finite extension of $\mathbb{Q}_p$ with no non-trivial unramified subextension, $f(x)$ be an irreducible monic polynomial in $\mathcal{O}_K[x]$, making $L=K[x]...

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### Would efficient factoring have any *other* useful applications?

This question is certainly somewhat opinion-based, but hopefully not hopelessly so.
The granddaddy of all applications for an efficient period finding or factoring capability (e.g. Shor's algorithm) ...

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### Fixpoints of $m\longmapsto \mathrm{rad}(\phi(m^2))$ under iteration

Given a strictly positive integer $m$ let $\alpha(m)=\mathrm{rad}(m\phi(m))$
be the radical (product of all distinct prime divisors) of the product of $m$ and of Euler's totient function $\phi(m)=m\...

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### Number of different factorizations

I define $\nu(n)$ the number of different factorizations for an integer $n$. I know there are papers about $\delta(n)$ the number of dividers for an integer $n$ (Landau, Euler, Dirichlet) but I still ...

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### Factoring higher-order differential operators

I have been researching various methods for solving differential equations. In particular, I want to better understand the factoring approach. For example, if we want to solve a general second order ...

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### Does having the discrete logarithm of prime factors of $n$ allow us to calculate any discrete log more efficiently?

Let $(p_1)^{k_1}(p_2)^{k_2}\dots$ be the prime factorization of $\varphi(n)$. Assuming that we have a value of order $(p_x)^{k_x}$ for all $x$, can we calculate the discrete log of any value in $\...

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### Solution to sixth order equation

I'm dealing with the expression $x = \frac{1}{3}y(y+1)(2y+1)^2(2y^2+2y+1)$. What is this approximately, if one is explicitly writing y in terms of x? There's no general formula for sixth powers ...

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### Groups suitable for algebraic group factorizations of integers

Quoting Wikipedia on Algebraic-group factorisation algorithm
Algebraic-group factorisation algorithms are algorithms for factoring an integer N by working in an algebraic group defined modulo N whose ...

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### Reducibility of $f(x)^{2^n}+1$ and $f(x)^{2^n}+g(x)^{2^n}$

Related to generalized Fermat numbers.
Let $f(x),g(x)$ be coprime polynomials with integer coefficients.
Assume that if $f(x)$ or $g(x)$ are of the form $h(x)^k$ then $k$ is power
of two.
Q1 Is it ...

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### Systematic approach to Weierstrass factorization

If you want to calculate the Taylor expansion of a function, you only need to know the derivatives of the function at the point of expansion.
Is there a similar algorithmic approach that can be ...

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### Integers with exactly three factor pairs whose successors are relatively prime

I am interested in the following problem, and will appreciate pointers around how it can be solved – partially or fully – and/or indicators around whether it is even tractable:
Characterize $N \in \...

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### Hensel's lemma, Bezout's identity, and the integers

Factorization in the ring $\mathbb{Z}[x]/(x^2+1)\mathbb{Z}[x]\cong \mathbb{Z}[i]$ is well known. For instance, $5$ and $13$ (and any prime $\equiv 1\pmod{4}$) are no longer prime.
The factorization ...

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### Factorization of argmax

We consider a function $f(s_{1:p}, a_{1:p})$, where $p>1$ is an integer, $s_{1:p}$ denotes $(s_1,\ldots,s_p)^\top \in R^p$, and $a_{1:p}$ denotes $(a_1,\ldots,a_p)^\top \in R^p$.
Question: What is ...

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### Positive divisors of $P(x,n)=1+x+x^2+ \cdots + x^n$ that are congruent to $1$ modulo $x$

This is a follow-up question to Positive integer solutions to the diophantine equation $(xz+1)(yz+1)=z^4+z^3 +z^2 +z+1$
Let \begin{equation}
P(x,n)= 1+x+x^2+ \cdots + x^n, \end{equation}
\begin{...

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### A diophantine equation involving partial sums of exponentials similar than the equation in Fermat's Last Theorem

I'm curious about the following diophantine equation from my invention: I don't know if this is in the literature, I wrote it using creativity in an attempt to write a variant of the equation in ...

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### $2$-adic valuation of Schur $P$-functions in the power-sum basis

For a partition $\lambda$, let $P_\lambda$ be the Schur $P$-functions (case $t=-1$ of Hall-Littlewood symmetric functions) and let $p_\lambda=p_{\lambda_1}p_{\lambda_1}\cdots p_{\lambda_k}$ be the ...

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### Next smooth number

I want to find the next $n \in \mathbb{N}$ such that
$$s < n = \prod_{p_i \in \mathbb{P}_B} {p_i}^{a_i}$$
Where $\mathbb{P}_B$ is the set of primes not greater than $B$
I know that we can generate ...

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### Need help interpreting this formula for the number of Goldbach partitions [closed]

1: Formula for the number of Goldbach partitions.
Let $g\left(n\right)$ denote the number of Goldbach partitions of even integer $2n$:
$$g_{\left(n\right)}=\sum_{3\leq p\leq2n-3}\left[\pi\left(2n-p\...

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### Functions over monoids which factor in two different ways

This is a follow-up question to this MO question, which was asked by Richard Stanley in a comment to my answer there.
Let $S$ be a commutative monoid and $f(x_1, \dots, x_n)$ be a function from $S^n$ ...

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### Numbers with large prime exponents and the ABC conjecture

By Fermat's Last Theorem, there are no solutions to the Diophantine equation $a^n + b^n = c^n$ for $a,b,c > 0$ and $n>2$. Beal's conjecture allows the exponents to be different (but also $>...