84 problems

Let $P\in \mathbb{Z}[t]$ be irreducible, and let $\mathfrak S_P$ denote the singular series … An integer with exactly two distinct prime divisors, called an $E_2$ integer, has the…

Koblitz's conjecture. There exists a constant $C>0$ such that

Bouniakowsky's conjecture. There are infinitely many primes of the form $P(z)$ for $z\in\mathbb{Z}^+$. This conjecture predicts that every integer polynomial satisfying the stated…

A prime is a positive integer greater than $1$ with no positive divisors other than $1$ and itself. Landau's conjecture. There are infinitely many primes of the form … This is one…

Let $f\in \mathbb Z[x]$ be an irreducible polynomial such that the integers in the set $f(\mathbb N)$ have no common factor other than $\pm 1$. Buniakowski's conjecture. The polyno…

Let $m\in\mathbb{N}$ and let $f_1,\ldots,f_m\in\mathbb{Z}[x]$ be polynomials. Suppose that there is no prime number $p$ such that … for every $n\in\mathbb{N}$. Schinzel–Sierpiński…

Let $f_n(x)$ be the polynomial sequence defined earlier in the paper, and let $a$ be a positive integer. Prime-value conjecture. For every positive integer $a$, there are infinitel…

Let $f\in\mathbb{Z}[x]$ be irreducible and have trivial fixed divisor. Bunyakowski's conjecture. The value $|f(n)|$ should be prime for infinitely many $n\in\mathbb{Z}$. This is a…

Bouniakowski–Schinzel conjecture. If $f(x)$ represents primes, then $f(x)$ has infinitely many prime values.

Primitive-root density conjecture. The quotient $R_g(f;x)$ tends to a limit as $x$ tends to infinity; denote this conjectural density by $\delta_g(f)$. This predicts that the relat…

Let $m\ge2$ be an integer, and let $f(X)\in\mathbb Z[X]$ represent infinitely many primes. A congruence class modulo $m$ is allowable if every integer $r$ in it satisfies…

Let $A$, $B$, and $C$ be relatively prime integers such that $A$ is positive, $A+B$ and $C$ are not both even, and $B^2-4AC$ is not a perfect square. Quadratic prime-values conject…

Finite-field Bateman–Horn conjecture. As $X\to\infty$ through powers of $q$,

Finite-field Bateman–Horn conjecture. As $X\to\infty$ through powers of $q$,

Extended Schinzel hypothesis H. If

Bateman–Horn conjecture. With

Let $\tau(n)$ denote the $n$th Fourier coefficient of Ramanujan's discriminant modular form. A prime occurs as a tau-value up to sign if $|\tau(n)|=\ell$ for some integer $n\geq 1$…

Sun's conjecture. Every integer $n>1$ has a decomposition $n=x+y$ with integers $x,y>0$ such that $x+ny$ and $x^2+ny^2$ are simultaneously prime.

Let $a,b\geqslant 2$ be integers, and choose maximal positive integers $k$ and $\ell$ such that $a=A^k$ and $b=B^\ell$. Granville–Pappalardi's power-base conjecture. There are infi…

Let $a,b\geqslant 2$ be integers. Say that there is no obstruction when neither $a$ nor $b$ is a power of an integer and there is no non-zero integer $Q$ such that … for all positi…

Bouniakovsky-type conjecture. A primitive, irreducible Dirichlet polynomial $f(s)$ with positive leading coefficient such that the set of values $f(\mathbb{Z}_{-})$ has no common d…

Prime-value conjecture for odd powers. (a) If $t\equiv 0\pmod 3$ and $d\neq 2,9$, then there are infinitely many primes of the form

Let $f$ be an irreducible polynomial of degree $g$, and let $n$ range over sufficiently large integers. Prime-values conjecture. There are infinitely many $n$ such that $f(n)$ is p…

Let $f(x,y)$ be a bivariate polynomial with non-negative coefficients and no constant term, and suppose that $f(x,y)$ does not represent any primes over the positive integers. For…

For $n\in\mathbb{Z}_+$, let $P^+(n^2+1)$ denote the greatest prime factor of $n^2+1$. The prime-values conjecture for $n^2+1$. There exist infinitely many primes of the form … This…