32 problems

Let $f_1,\ldots,f_k$ be irreducible integer polynomials satisfying the admissibility condition. Let $P(x)$ denote the number of integers $n\leq x$ for which all $f_i(n)$ are prime.…

Bouniakowsky's conjecture. Then $f(n)$ is prime for infinitely many positive integer values of $n$. This is the motivating conjecture for studying eventually prime-free sequences,…

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

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…

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…

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

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…

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 $f_n(x)$ 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 $f(t)\in\mathbb{Z}[t]$ be a separable polynomial of degree $\operatorname{deg} f\geq 1$. Let $\mu$ denote the Möbius function, let $\rho_f(m)$ denote the number of residue clas…

Consider the four cases (a)–(d) for a prime $p$, in which the relevant factors of $p-1$ and $p+1$ are required to be prime; equivalently, each case gives a triple of linear polynom…

Generalized Landau conjecture. The polynomial $f$ represents infinitely many primes.

Let … where $\left(\frac{-1}{p}\right)$ is the Legendre symbol. $n^2+1$ prime-value conjecture. The expected number of primes of the form $n^2+1$ less than or equal to $N$ is … Thi…

Let $c_n$ denote the coefficients of the $j$-function considered in the paper. Prime-value conjecture. There are infinitely many $n$ such that $c_n$ is prime. The computation finds…

Let $p$ be a prime, set $q=p^3$, and define … The cubic-prime-power projective-prime conjecture. There are infinitely many values of $q=p^3$ such that $m=1+q+q^2$ is prime. The con…

Fix a prime integer $n\ge 3$, and let $p$ range over prime numbers. Define … The fixed-exponent projective-prime conjecture. For every fixed prime $n\ge 3$, there are infinitely ma…

Let $n\ge 2$ be an integer and let $q\ge 2$ be a prime power. Define the projective degree … A projective prime is a prime value of $m$. The projective-prime infinitude conjecture.…

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

Let $f_{1},f_{2},\cdots,f_{M}\in\mathbb{Z}[x]$ be distinct irreducible polynomials with positive leading coefficients. Define … Put $f(n)=\prod_{i=1}^{M}f_{i}(n)$, suppose that…

Let $A/\mathbb{Q}$ be an abelian variety satisfying the hypothesis $(\operatorname{Triv}_A)$, and suppose that $C_A\neq 0$. Koblitz's conjecture for abelian varieties. … In particu…

Multivariable Bateman–Horn conjecture. As $P\to+\infty$,

Let $f(x)\in\mathbb{Z}[x]$ be a quadratic irreducible polynomial such that the sequence $\\{f(n)\\}_{n\geq 1}$ does not have a common factor. Buniakowski's conjecture. There exist…