All Questions
Tagged with analytic-number-theory polynomials
7 questions
17
votes
3
answers
2k
views
About the prime divisors of values of polynomials
Let $P$ be a polynomial having integer coefficients (and degree $\geq 3$), and let $\mathscr P_P$ be the set of prime numbers dividing some value $P(n)$ with $n \in \mathbb Z$.
Is it true that $\...
- 3,866
9
votes
0
answers
324
views
Semi-primes represented by quadratic polynomials
According to Lemke-Oliver, irreducible quadratic polynomials $G$ with positive leading coefficient and $\rho(2)<2$, (where $\rho(m)$ denotes the number of incongruent solutions to the congruence $G(...
- 195
7
votes
2
answers
997
views
Upper density of the set of $n$'s such that $p(n)$ is prime, where $p$ is polynomial
The starting point for this question is the following (false) statement
$\forall n\in \mathbb{N} (n^2 + n + 41 \text{ is prime}).$
Given a polynomial function $p:\mathbb{N} \to \mathbb{N}$ ...
- 48.9k
6
votes
1
answer
665
views
On the distribution of roots modulo primes of an integral polynomial
For motivation and related questions, see below.
Rough sketch of the question.
View $\bigsqcup_{p \text{ prime}} (\mathbb{Z}/p\mathbb{Z})$ as a ‘subset’ of the unit circle, via $a\pmod{p} \mapsto e^{...
- 83
1
vote
0
answers
244
views
Möbius function and polynomials
Let $\mu$ be the Möbius function. It is well known that $\sum_{n|k} \mu(n) = 0$ for $k>1$. What could be said about the polynomials $R_k = \sum_{n|k} \mu(n) x^n$ for $x \in [0,1]$? There does not ...
- 433
0
votes
1
answer
356
views
A sufficient condition for a set of primes to be the set of reducibility of an integer polynomial
Let $P$ be the set of all positive primes. Let $S$ an arbitrary infinite subset of $P$ satisfying the following assumption: there exists a finite Galois extension $K$ of $\mathbb{Q}$ and a conjugacy ...
user145520
0
votes
1
answer
137
views
A density zero set of primes dividing the values of a non-constant integer polynomial
For a given $P\in \mathbb{Z}[x]$ call a positive prime $p$ good if there exists $n\in \mathbb{Z}$ such that $p$ divides $P(n)$. Does there exist a non-constant $P$ such that the set of good primes has ...
user145520