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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 $\...
Konstantinos Gaitanas's user avatar
15 votes
1 answer
495 views

Is the number of representations as the sum of two elements of a polynomial sequence always small?

Let $f(x) \in \mathbb{Z}[x]$ be a degree $d>1$ polynomial with integer coefficients. Define $$r(n) := | \{x,y \in \mathbb{Z} : f(x)+f(y) = n \}|. $$ My question is: Is it true that $r(n)...
Mark Lewko's user avatar
13 votes
3 answers
949 views

Polynomials vanishing modulo some integer $n$

It is well-known that a polynomial $q \in \mathbb Z[t]$ vanishes modulo $p$ only if it lies in the ideal $J_p$ generated by $p$ and $t^p-t$. This means that either the degree is large (at least $p$) ...
Andreas Thom's user avatar
  • 25.5k
12 votes
2 answers
902 views

Infinitely many irreducible polynomials of the form f(X^2) + X mod 3?

Are there infinitely many polynomials $f \in \mathbb{F}_3[X]$ for which $f(X^2) + X$ is irreducible?
Pablo's user avatar
  • 11.3k
12 votes
1 answer
1k 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 $p$...
Joël's user avatar
  • 26k
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(...
Delmastro's user avatar
  • 195
8 votes
1 answer
1k views

An elementary lower bound on the number of primes

Recall the second Chebyshev function: $$\psi(x) = \sum_{p \leq x} \lfloor \log_p x \rfloor \log p$$ where $x$ is a positive integer, and $p$ runs over all primes $\leq x$. In a hunt for an "...
Pablo's user avatar
  • 11.3k
8 votes
0 answers
224 views

Is there an approximate formula for the discriminant of a sparse polynomial?

Consider integer polynomials $P \in \mathbb{Z}[X] \setminus \{0\}$ of a degree $D \geq 1$ and without multiple complex roots. Let me introduce a notation $$ d(P) := \frac{1}{D} \log{|\mathrm{Disc}(P)|}...
Vesselin Dimitrov's user avatar
7 votes
2 answers
998 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 $p:\mathbb{N} \to \mathbb{N}$ ...
Dominic van der Zypen's user avatar
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^{...
user98708's user avatar
5 votes
1 answer
392 views

Divergence of primes dividing polynomials

Let $Q : \mathbb{Z} \rightarrow \mathbb{Z}$ be a polynomial. Form the set $$M_{Q} := \{p:\text{ }p\text{ is prime, }\exists n_{p}\in \mathbb{Z}\text{ so that }p|Q(n_{p})\}$$ Is $$\sum_{s \in M_{Q}}\...
Siddharth Iyer's user avatar
5 votes
1 answer
430 views

How many roots of polynomial in $\mathbb Z[x]$ and $\mathbb Q[x]$ are integers on average?

Given $d,B>0$ the number of polynomials in $\mathbb Z[x]$ of degree $d$ and coefficient size at most $B$ have at least one integer roots should be $B^{O(d)}f(d)$ at some function $f$ (from Random ...
Turbo's user avatar
  • 13.9k
5 votes
0 answers
205 views

Is there a polynomial version of Wilson's theorem which can avoid Cramer flavored conjectures?

Wilson's theorem states that a natural number $n > 1$ is a prime number if and only if the product of all the positive integers less than $n$ is one less than a multiple of $n$. Is there a version ...
Turbo's user avatar
  • 13.9k
4 votes
1 answer
271 views

The highest power of $2$ dividing a polynomial evaluated at $x=3$

Let $\nu_2(a)$ be the $2$-adic valuation of an integer $x$, i.e. the largest power $t$ such that $2^t$ divides $x$. Define the operator $D=x\frac{d}{dx}$ and the polynomial $\Phi_k(x)=\frac{x^{k+1}-1}{...
T. Amdeberhan's user avatar
4 votes
0 answers
134 views

$\delta$-equidistributed polynomials over finite fields

I'm trying to show that a polynomial over finite (prime) field is "close enough" to being equidistributed over its range. A polynomial $p(\cdot)$ from $\mathbb{F}^n$ to $\mathbb{F}$ is $\...
GWB's user avatar
  • 301
4 votes
0 answers
432 views

square-free parts of values of polynomials

Given a polynomial $f(x) \in \mathbb{Z}[x]$ of degree $d$, consider the following three sets: $$N_1(x) = \#\{k \leq x: f(k) \text{ is square-free}\}$$ $$N_2(x) = \#\{n \leq x: n = f(k) \text{ is ...
stl's user avatar
  • 585
3 votes
1 answer
372 views

How many ways can $N$ be written as a sum of terms in the form $2^i3^j$?

Given a positive integer $N$, let $f(N)$ be the number of ways $N$ can be decomposed as a sum of terms of the form $2^i3^j$, where each such term appears at most once in the sum. For example, $f(10) = ...
Gautam's user avatar
  • 1,703
3 votes
2 answers
365 views

Largest prime factors of integer polynomials

I have a question in analytic number theory which is closely related to the open problem (Bunyakovsky conjecture and more generally, Schinzel's hypothesis H) that asks you if, any irreducible ...
James Moriarty's user avatar
3 votes
1 answer
194 views

Divergence of a series related to Schinzel's hypothesis H

The Series Consider the series identity $$\Phi(s) = \sum_{n=1}^\infty \frac{\mu(n) (\log n)^k}{n^s} \sum_{r \in R(n)} \zeta(s,r/n) = \sum_{n=1}^\infty \frac{\Lambda_k'(f(n))}{n^s}$$ $$R(n) = \left\...
Liam Eagen's user avatar
3 votes
1 answer
344 views

Number of prime factors of a polynomial discriminant

Let $f(x)\in\mathbb{Z}[x]$ be a polynomial of degree $d$ and naive height (maximum of the absolute values of the coefficients) at most $H$. Is there anything known about the number of prime factors of ...
Nicolas Banks's user avatar
2 votes
2 answers
370 views

Link between Irreducible Factors and Prime Factors (or Cycles of a Permutation)

In "Anatomy of Integers and Permutations", http://www.dms.umontreal.ca/~andrew/PDF/Anatomy.pdf, Granville gives a calibration of cycles of a permutation and prime factors of an integer. "We know ...
The Substitute's user avatar
2 votes
2 answers
208 views

On the number of values with exactly $k$ prime factors of a given polynomial

This is surely be a well studied problem. Let $f(x) \in \mathbb{Z}[x]$. Is there some $k \in \mathbb{N}$ such that there are infinitely many $n \in \mathbb{Z}$ where $f(n)$ has exactly $k$ prime ...
Paul Cusson's user avatar
  • 1,763
2 votes
0 answers
159 views

Large prime divisors of values of a polynomial, in a given residue class

Let $f(X) \in \mathbb{Z}[X]$ be an irreducible polynomial of degree $d \geq 2$. Let $q \in \mathbb{N}$ be an integer, and let $q \mathbb{Z} + r$ be a residue class that contains infinitely many primes ...
Jakub Konieczny's user avatar
2 votes
0 answers
84 views

quadratic residues and cubic polynomials [closed]

I'm really not sure about this, but I've heard somewhere that for any prime $p$, $|\sum_{x=0}^{p-1} (\frac{ax^3 +bx^2 +cx +d}{p} ) |\le \sqrt{2p}$ holds. Does anyone know a proof for this inequality ...
Junsukim's user avatar
  • 141
2 votes
0 answers
67 views

Density of integral values of a rational function

Let $\mathbf{x} = (x_1, \cdots, x_n)$, and consider a rational function $F : \mathbb{R}^n \rightarrow \mathbb{R}$ be given by $$\displaystyle F(\mathbf{x}) = \sum_{i = 1}^m \frac{Q_i(x_1, \cdots, x_{...
Stanley Yao Xiao's user avatar
2 votes
0 answers
121 views

How to choose a prime p s.t. n-th cyclotomic polynomial splits into as much as possible irreducible polynomials while p is almost constant size?

The reason I ask this question is that cyclotomic polynomial is critical to the construction of lattice-based cryptography. In most of the existing lattice-based cryptographic schemes, $n$ is usually ...
user67451's user avatar
1 vote
0 answers
323 views

On fifth powers forming a Sidon set

We call a set of natural numbers $\mathcal S$ to be a Sidon Set if $a+b=c+d$ for $a,b,c,d\in \mathcal S$ implies $\{a,b\}=\{c,d\}$. In other words, all pairwise sums are distinct. Erdős conjectured ...
Sayan Dutta's user avatar
1 vote
0 answers
88 views

Distribution of number of integer solutions in box to bivariate polynomials?

Take a bivariate polynomial of degree $d_x+d_y>\max(d_x,d_y)>1$ in $\mathbb Z[x,y]$ with coefficients bound in absolute value by $b$ ($d_x$ is $x$-degree and $d_y$ is $y$-degree). What is the ...
Turbo's user avatar
  • 13.9k
1 vote
1 answer
189 views

Explicit bivariate quadratic polynomials where Coppersmith is better than standard solver?

http://www.numbertheory.org/pdfs/general_quadratic_solution.pdf gives a general method to solve quadratic bivariate diophantine equation while Coppersmith introduced a method to solve bivariate ...
Turbo's user avatar
  • 13.9k
1 vote
0 answers
276 views

Method of Coppersmith optimal for multivariate?

It is shown that Coppersmith method yields optimal integer root extraction for univariate polynomials in https://arxiv.org/abs/1605.08065 and a follow up work attempts this for bivariate polynomials ...
Turbo's user avatar
  • 13.9k
1 vote
0 answers
202 views

Prime generating polynomials

Continuation to this previous question. According to Lemke-Oliver, an irreducible polynomial $G$ of degree $g$ with positive leading coefficient and $\Gamma_G\neq0$ (with $\Gamma_G$ a certain factor ...
Delmastro's user avatar
  • 195
0 votes
1 answer
431 views

Reason Coppersmith fails here?

Take classic problem of finding $P,Q$ in balanced semi-prime $N=PQ$. $P$ has a binary expansion and so does $Q$. We can set the binary $0/1$ variables to be $x_1$ through $x_{\lceil\log P\rceil}$ and $...
Turbo's user avatar
  • 13.9k
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 ...
user avatar
0 votes
1 answer
159 views

Parity and number of squares taken by polynomials in a range?

I have a polynomial $f(x)=a^2x^2+bx+c\in\mathbb Z[x]$ with $f(x)$ not a constant times a square and $abc\neq0$ and I want to know how many $x$ between $-a$ and $a$ the polynomial is a perfect square. ...
Turbo's user avatar
  • 13.9k
0 votes
0 answers
115 views

Maximum number of integer solutions with some size constraints to bivariate polynomials?

Take a bivariate polynomial of total degree $d$ satisfying $d=d_x=d_y>1$ in $\mathbb Z[x,y]$ with coefficients bound in absolute value by $b$ ($d_x$ is $x$-degree and $d_y$ is $y$-degree). Given a ...
Turbo's user avatar
  • 13.9k
0 votes
0 answers
152 views

On the $\mathsf{LCM}$ of a set of integers defined by moduli of powers

For integers $a,b,t$ define $$\mathcal R_t(a,b)=\{q\in\mathbb Z\cap[1,\min(a^t,b^t)]: a^t\equiv b^t\bmod q\}$$ and $\mathsf{LCM}(\mathcal R_t(a,b))$ to be $\mathsf{LCM}$ of all entries in $\mathcal ...
VS.'s user avatar
  • 1,826
-2 votes
1 answer
181 views

Polynomials of minimum degree that interpolate primes in intervals

Given an interval $[a,b]$ what is the minimum degree of univariate polynomials in $\mathbb Q[x]$ that passes through all primes between $a$ and $b$ (denoted by $\mathbb P[a,b]$ with total number of ...
VS.'s user avatar
  • 1,826