Questions tagged [computational-number-theory]

For years, there was a simple reason why the largest known prime is of the form $2^{p}-1$: We had the Lucas-Lehmer test which was specific to Mersenne numbers, and faster than all other known methods. ...

Let $p$ be a prime with $p \gt 3$. Consider the polynomial $f = x^3 - 3x -1$. Suppose $f$ is irreducible over $\mathbb{F}_{p}$. Let $E$ be the splitting field of $f$ over $\mathbb{F}_{p}$, and let $\...

Given a number field $K$, I would like to compute (in Sage) generators for the group of totally positive units of $K$. Update: I've tried some code (details below), which I've received some help on in ...

If we take ceil and floor of $\sqrt{x(x+1)}$ (when it exists) we get $x$ and $x+1$ respectively. Is there an analog of this assuming roots exist in modular arithmetic (at least modulo primes)? ...

Is it any easy to factor $p-1$ when $p$ is a prime compared to general factorization problem? What about when $2p+1$ is also a prime?

Let $L$ be a cyclotomic field, and $P$ a prime ideal of $\mathcal{O}_L$. is there any symbol for the equation $ax^2 + by^2 =1 \mod P$ and if so, is it computable in polynomial time? if $a$ is ...