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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Primality testing by reversible computation using the prime number theorem

Suppose we want to build a primality testing algorithm for the numbers limited to the set $A =\{1, ..., 2^n\}$ and $n$ is reasonably large. The prime-number theorem tells us that there are ...
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Coefficients of 0,1-polynomials factorization

Let $n\in\mathbb{N}$ and $P_{n}$ is $0,1$-polynomial whose coefficients are binary digits of $n$. Let $Q_{1}(x) \cdot Q_{2}(x) \cdots Q_{m}(x)$ - polynomial factorization (over integers) of $P_{n}$. ...
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Reference Request: Factorization method for polynomials whose maximum absolute value of coefficient is 1

So, today I came up with a method for factoring polynomials whose coefficients are either $-1$ or $1.$ Let me explain with examples. Example No. 1. Factorize $P(x)=x^8+x^7+1$ Solution. It is known ...
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Being interested in these polynomials, would like to clarify one small observation. Let $n\in\mathbb{N}$ and $P_{n}$ is $0,1$-polynomial whose coefficients are binary digits of $n$. Let $n$ has prime ...
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On the multiplicative group of quotients of polynomial rings

Related to this. The $p+1$ factorization algorithm works over $\mathbb{Z}/n\mathbb{Z}[x]/f(x)$ and hopes $p+1$ to be smooth. We are trying to generalize this to multivariate case and also try to find ...
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Evaluating the generalized continued fraction obtained from the factorization of a bivariate polynomial equation

Happy New Year, MO community! We need someone expert in Generalized Continued Fractions (GCFs), with a deep knowledge of the GCFs’ convergence properties, to solve the following problem. PROBLEM ...
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Has anyone studied factoring as a CO-product?

In factorization, like integer factorization, you start with an integer and end up with a kind-of list of pairs of other elements, namely the factors. I want to explore the "Co-ness" of this....
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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?
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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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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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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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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 ...