4
votes
2answers
281 views

Can a polynomial be almost always divisible by a member of a finite set of primes?

Special case of Bunyakovsky conjecture Let $f(x)$ be non-constant irreducible polynomial with integer coefficients, no fixed prime factor and positive leading coefficient. Let $S$ be a finite set of ...
4
votes
3answers
307 views

Set of primes dividing polynomials and composition

For a non-constant polynomial $A \in \mathbb{Z}[x]$, let $\mathcal{P}(A)$ denote the set of prime numbers $p$ which divide $A(n)$ for some integer $n$. If $\mathcal{P}(A) \subseteq \mathcal{P}(B)$ for ...
8
votes
1answer
331 views

Least prime $p$ such that an irreducible polynomial of degree $n$ has no root modulo $p$?

This question is inspired by an old question of Greg Kuperberg, about how small is the first prime $p$ which makes a given monic polynomial $P$ with integral coefficient have a (simple) root modulo ...
30
votes
2answers
1k views

A question on maps from $\mathbb{Z}/p\mathbb{Z}$ to itself

Let $p\geq 3$ be a prime number, and let $u:\mathbb{Z}/p\mathbb{Z}\to \mathbb{Z}/p\mathbb{Z}$ be a map such that, for all $l\in \mathbb{Z}/p\mathbb{Z}$,$l\neq 0$, the map $k\mapsto u(k+l)-u(k)$ is a ...
0
votes
2answers
451 views

On reducible polynomials with positive coefficients, $1$ as constant coefficient and certain bounds on coefficients

Given $a \in \mathbb{Z}$ with $a > 1$. Let $g(x) \in \mathbb{Z}[x]$ be a polynomial with $g(a)=\pm 1$. Let $h(x) \in \mathbb{Z}[x]$ be a polynomial with $h(a)= p$, a prime. Let $g(x)$ and $h(x)$ ...
2
votes
1answer
234 views

10 factors for x^2 coefficient in quadratic sieve?

I wrote a quadratic sieve and I tried plugging in all the same parameters as the wikipedia article says msieve uses: http://en.wikipedia.org/wiki/Quadratic_sieve#Parameters_from_realistic_example It ...
15
votes
1answer
1k views

Primes represented by two-variable quadratic polynomials

I'm looking over a paper, "Primes represented by quadratic polynomials in two variables" [1] which attempts to characterize the density of the primes in two-variable quadratic polynomials. Its ...
0
votes
0answers
650 views

Coprime polynomials

Consider the set of polynomials (in standard form) P: the coefficients in the polynomials are integers. Quite like Z (integers), every distinct member of P has a unique 'prime factorization'. ...
1
vote
3answers
837 views

solutions to equation mod a prime

I know that characterizing the solutions to an equation in a finite field is generally difficult, but I was wondering if anyone had anything to say about the equation (ab)^2 + a^2 + b^2 = 0 mod p I ...
30
votes
1answer
3k views

Integers not represented by $ 2 x^2 + x y + 3 y^2 + z^3 - z $

EDIT, 9 March 2014: when I asked this in 2010, I did not have the courage of my convictions, and so did not ask for an if and only if proof, as Kevin Buzzard quite properly pointed out. Such problems ...
8
votes
4answers
1k views

What primes divide the discriminant of a polynomial?

Given a monic polynomial $p(t) = t^n + ... + c_1 t + c_0$ with integer (or rational) coefficients and with roots $a_1, \dots a_n$, we can compute its discriminant, which is defined to be $\prod_{i< ...