# Tagged Questions

For questions about finding factors of e.g. integers or polynomials

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### Under what conditions on $f$, is $f(az)=g(a)f(z)$?

Formal Statement Given nonzero constant $a \in \mathbb{C}$, $|a|>0$ and $f:\mathbb{C} \to \mathbb{C}$, under what conditions on $f$ does the following hold? f\left(a z\...
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### On factoring given $PQ-1$ has small factors.

Suppose we have an RSA number $PQ$ where $PQ-1$ has small factors. Will this give any advantage to factor $PQ$?
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### How to factor $a^3 - b^3$?

I know the answer is $(a - b)(a^2 + ab + b^2)$, but how do I arrive there? The example in the book I'm following somehow broke down $a^3 - b^3$ into $a^3 - (a^2)b + (a^2)b - a(b^2) + a(b^2) - b^3$ and ...
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### Product of heights of factors smaller than length of a polynomial with integer coefficients

I have the following question. Given a (univariate) polynomial with integer coefficients, I want to prove, if true, that the product of heights of its (irreducible) factors is smaller or equal to its ...
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### Expanding an infinite product of infinite series

Here's a fragment of something I posted in an answer a few months back: \begin{align} & \left( 1 + \frac 1 {a_1} + \frac 1 {a_1^2} + \frac 1 {a_1^3} + \cdots \right) \\ \times {} & \left( ...
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### About an integer factoring algorithm

I have been toying with the following algorithm: ...
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### Separable polynomials are the product of the minimal polynomials of their roots?

I see the following claim in this answer: Since $f$ is separable, it follows that $f(x)$ must be the product of minimal polynomials of [its roots] But, I don't know how we justify this claim. ...
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### Properties of integer matrices $A$ such that $A^{p}=I$ for $p$ a prime integer.

Problem Statement: Let $p$ be an integer prime, and let $A$ be an $n\times n$ integer matrix such that $A^{p} = I$ but $A \neq I$. Prove that $n \geq p − 1$. We have been learning factoring of ...
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### Factorization of vectors $(Z^T Z)^n Z^T-(X^T X)^n X^T$

Is there a way to factorize expression $(Z^T Z)^k Z^T-(X^T X)^k X^T$ where $Z$ and $X$ are real column vectors in $\mathbb{R}^n$, such that \begin{align} (Z^T Z)^k Z^T-(X^T X)^k X^T= (Z-X)^T P(Z,X) ...
I need help to factorise the following polynomial: $x^4 - 2x^3 + 8x^2 - 14x + 7$ The solution I need to reach is $(x-1)(x^3 - x^2 + 7x - 7)$. I need to factorize to this exactly as it is for a ...
On pp 255 - 256 (footnote 7) of "Love & Math", Edward Frenkel states that we can factor a quadratic in terms of its solutions $x_1$ and $x_2$ as: $ax^2 + bx + c = a(x - x_1)(x - x_2)$ Where does ...