22 problems

Let $f\in\mathbb Z[x]$ be a polynomial that does not have the form $c(g(x))^2$. Define … where $\Omega:\mathbb Q^\times\to\mathbb Z$ is the group homomorphism satisfying…

One-variable Chowla conjecture. The sequence $\lambda(f(n))$ has average value $0$ as $n$ ranges over any arithmetic progression. Equivalently, for every arithmetic progression…

Squarefree-value conjecture. The number of integers $n$ with $1\leq n\leq x$ for which $f(n)$ is squarefree should satisfy

Infinitely-many-primes conjecture for $n^2+1$. There exist infinitely many primes of the form $n^2+1$.

Let $F(x)\in\mathbb{Z}[x]$, and let $\Psi_F^*(x,y)$ denote the counting function for the relevant integers whose values under $F$ satisfy the smoothness condition encoded by the so…

Let $f(t)\in\mathbb{Z}[t]$ have distinct irreducible factors over $\mathbb{Z}[t]$ of degrees $d_1,\ldots,d_g\geq 1$. Let $\Psi_f(x,y)$ denote the number of integers $n\in[1,x]$ suc…

Verstraëte's dichotomy conjecture. For some constant $\rho=\rho(k,f)$ depending only on $k$ and $f$, the maximal size of such a set $A$ is either

Let $q=p^k$ be a prime power, and let $f\in\mathbb{F}_q[T][X]$ be a fixed irreducible polynomial with $X$-degree $d\geq 2$. Define … Let $V_f:=\{g\in\mathbb{F}_q[T]:f(X+g)=f(X)\}$…

Let $B$ be the smoothness bound in estimates for the distribution of smooth values of polynomials, and consider the range of $B$ for which such estimates are sought. Granville's co…

Let an SP number be a number in the class considered in the paper, and let $x$ range over positive integers. Bunyakovsky's conjecture for SP numbers. There are infinitely many SP n…

Let $F(t)\in\mathbb{Z}[t]$ be irreducible, and let $H_F(x,y,z)$ count the integers $n\leqslant x$ for which $F(n)$ has a divisor in $(y,z]$. Let $g$ and ${\mathcal E}$ be the param…

Maximal subword-occurrence conjecture. The same equality should hold for any finite word $w$:

Let $f\in\mathbb{Z}[x_1,\ldots,x_h]$ be primitive and squarefree, and let $\mathbb{Z}^h(X)=\{{\rm v}\in\mathbb{Z}^h:|{\rm v}|\leq X\}$. Squarefree conjecture, alternative version.…

Let $h\in\{1,2\}$, let $f\in\mathbb{Z}[x_1,\ldots,x_h]$ be primitive and squarefree, and let $\mathbb{Z}^h(X)=\{{\rm v}\in\mathbb{Z}^h:|{\rm v}|\leq X\}$. Define…

For a polynomial $P\in{\mathbb Z}[x]$, let $\Psi_P(x,y)$ count the integers among $\{P(n):1\leq n<x\}$ whose prime factors are all at most $y$. Weak smooth-values conjecture. For e…

Let $P\in{\mathbb Z}[x]$ be an irreducible polynomial of degree $d$, and let $\Psi_P(x,y)$ be the number of integers in the set $\{P(n):1\leq n<x\}$ whose prime factors are all at…

Let $p=n^4+1$ with $n\geq 1$, and let $\Lambda$ denote the von Mangoldt function. Quartic prime-values conjecture. The expected number of quartic primes $p=n^4+1\leq x$ has the asy…

For integers $n\geq 1$, consider the pair of polynomial values $n^4+1$ and $n^4+3$. Quartic twin primes conjecture. There are infinitely many twin primes $n^4+1,n^4+3$ as…

Let $X$ be sufficiently large, let $h$ and $\delta$ be parameters, and let $T(h,\delta)$ denote the number of integers $n\in (X,2X]$ for which $n^3+2$ has at least $h$ prime factor…

Squarefree-value residue class conjecture. For all but finitely many primes $p$, the set $\mathcal S(f)$ contains infinitely many elements from every nonzero residue class modulo…

Let $f_1(x),\ldots,f_r(x)\in\mathbb{Z}[x]$ and let $f(x)=f_1(x)\cdots f_r(x)$. Assume that the $f_i$ are distinct and irreducible in $\mathbb{Z}[x]$, have positive leading coeffici…

Let $f(x)=adx^d+dots+a0 i mathbb{Z}[x]$ be an irreducible polynomial of degree $d a 3$, with $ad0$, and let $psif(n)$ denote the least common multiple of the values of $f$ up to…