In "On computing factors of cyclotomic polynomials", Richard P. Brent gives the identity $$ 4 \Phi_n(x) = A_n(x)^2 - (-1)^{(n-1)/2} n B_n(x)^2 \qquad (1) $$ where $n$ is odd squa …

I read somewhere that 3SAT can be used to solve Integer Factorization. If that is true, could someone teach me a simple example of generating the 3SAT by using a small number? Let …

I've been running a search for Mordell curves of rank >=8 for about 12 months and have identified approximately 280,000 curves in our archivable range, amongst many millions that a …

Given an integer d, is there a way of finding max(n^m), where d=n*m, except brute-forcing it ? n and m don't have to be primes. For example: The number 12 can be factorized into …

Background: In the 1700s, Euler solved the Basel Problem, which was to solve $\sum_{n=1}^\infty\frac{1}{n^2}$ in closed-form. Euler showed that it was equal to $\frac{\pi^2}{6}$ b …

A basic principle in complex function theory is that one can split off zeros of holomorphic functions in a similar way as for polynomials: If $f$ is holomorphic near $0$ and $f(0) …

I am working with elliptic curves defined over $\mathbb{Q}(t)$, my current task is to search is to search torsion-free points in these curves after possibly changing $t$ by a ratio …

Let $N$ be a positive integer. Thanks to the Miller-Rabin test and the work of Agrawal, Kayal and Saxena, these days people have much much faster algorithms for testing whether $N$ …

I apologize if this is a naive question about greatest prime factors (gpf). I was thinking about the sequence of integers where $gpf(x) \le p$ where $p$ is any prime. Clearly, as …

While reading Peter Shor's paper Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer, I came across the following quote: "This sche …

Let a gaussian circle $C_R$ be any circle defined by the equation: $$x^2+y^2 = R, (x,y) \in \mathbb{R}^2$$, where $R$ is the norm of a gaussian integer ($R=a^2+b^2, (a,b) \in \mat …

Some power series factorize; $1+\sum_{n=1}^\infty x^n=\prod_{n=1}^\infty (1+x^{2^n})$ and $1+\sum_{n=1}^\infty x^{2n}/(2n+1)!=\prod_{x=1}^\infty (1+x^2/n^2\pi^2)$ for example; whi …

Suppose you need to slow down a turning motor so that a gear turns at an angular velocity $\frac{a}{b}$ of that of the motor shaft, where $a$ and $b$ are natural numbers. For exam …

Hi everyone, I want to build the following factor analysis model and I've samples observations of data (i.e., $m(s)$), labeled with factor $s$. The observations are affected by bo …

I have two questions about the class of integer-coefficient polynomials all of whose roots are rational. I asked this at MSE, but it attracted little interest (perhaps because it i …