# 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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### 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?
1 vote
57 views

### 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?
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
105 views

### 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 ...
1 vote
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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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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 $$N_{\delta}(B)=\#\{(x_1,\dots,x_n) ... 2 votes 0 answers 47 views ### A gsvd variation: Two SVDs with common matrix The generalized singular value decomposition (gsvd) is described on wikipedia here. Since there are several conventions, I'll just briefly present one. The "MATLAB" convention decomposes two ... 1 vote 0 answers 151 views ### Construct special "joint SVD" from separate SVDs Given two matrices, A,B\in\mathbb{C}^{n\times n} which can be written as$$ 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\... 3 votes 0 answers 123 views ### 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 (... 1 vote 1 answer 120 views ### 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 ... 0 votes 0 answers 161 views ### 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)... 1 vote 0 answers 88 views ### 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. ... 2 votes 1 answer 298 views ### 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]... 12 votes 1 answer 555 views ### 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) ... 10 votes 1 answer 296 views ### 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\... 0 votes 0 answers 34 views ### 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 ... 3 votes 1 answer 139 views ### 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 ... 1 vote 2 answers 324 views ### 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 \... 6 votes 1 answer 903 views ### 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 ... 4 votes 1 answer 176 views ### 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 ... 5 votes 0 answers 144 views ### 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 ... 1 vote 0 answers 40 views ### 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 ... 3 votes 0 answers 128 views ### 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 \... 9 votes 1 answer 604 views ### 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 ... 1 vote 0 answers 133 views ### 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 ... 2 votes 1 answer 333 views ### 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{... 0 votes 0 answers 136 views ### 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 ... 10 votes 1 answer 245 views ### 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 ... 3 votes 0 answers 93 views ### 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}$$Where \mathbb{P}_B is the set of primes not greater than B I know that we can generate ... 1 vote 0 answers 123 views ### Need help interpreting this formula for the number of Goldbach partitions [closed] 1: Formula for the number of Goldbach partitions. Let g\left(n\right) denote the number of Goldbach partitions of even integer 2n:$$g_{\left(n\right)}=\sum_{3\leq p\leq2n-3}\left[\pi\left(2n-p\...
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 $S^n$ ...
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 \$>...