63 problems

Let $f$ be a quadratic polynomial defined over ${\mathbb Q}$, and let a rational periodic point have exact period $N$. Poonen's conjecture. Such a polynomial cannot have a rational…

Flynn–Poonen–Schaefer conjecture. There is no quadratic polynomial $\varphi_{2,c}(z)=z^2+c\in\mathbb Q[z]$ with a rational point of exact period $n$.

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…

For a quadratic polynomial, let its core entropy be the entropy of the dynamics on its Hubbard tree, and let the polynomial vary in the space of quadratic polynomials equipped with…

Amdeberhan–Medina–Moll conjecture. The value $P_{1,1}(n)$ is not a square for $n\geq 4$, and $P_{4,1}(n)$ is not a square for $n\geq 1$.

Baker–DeMarco's conjecture. The parameters $c\in\mathbb{C}$ for which both $0$ and $1$ are preperiodic for $f_c$ are exactly

Let $n$ be a positive integer, and let $\pi_{f_2}(x)$ denote the counting function for integers $n$ such that $n^2+1$ and $n^2+3$ are both prime. For each prime $p\geq 5$, let…

Let $f(z)=z^2+c$ be a quadratic polynomial whose Julia set is full, and let $[-\beta,\beta]$ denote its spine. The critical value is $c$. Spine conjecture. The critical value belon…

Secondary limb conjecture. The secondary limb condition implies a priori bounds.

Let $f(z)=z^2+c$ be a complex quadratic map satisfying the Collet–Eckmann condition, and let the basin of infinity be the set of points whose iterates under $f$ tend to infinity. A…

Let $N$ be an integer with $N\geq 4$, and let $f(z)\in\mathbb{Q}[z]$ be a quadratic polynomial. A point is of exact period $N$ if its forward orbit under $f$ has least period $N$.…

Let $P_c(z)=z^2+c$ be a quadratic polynomial, and call $P_c$ hyperbolic if it has an attractive or super-attractive periodic point in the complex plane. Define … Let $\mathcal{M}$…

Let $f_a$ denote a real quadratic polynomial in the real quadratic family, and let its kneading sequence be the associated kneading invariant. A parameter has an attractive periodi…

Consider the quadratic family in the chosen normal form, and let $Q$ be the set of parameters for which the corresponding filled Julia set contains an interior point. Let $M$ be th…

Let $f\in\mathbb{Z}[x]$ be a quadratic polynomial. Call $f$ stable if every iterate $f^n$ is irreducible over $\mathbb{Q}$, and call $f$ critically infinite if the forward orbit of…

Let $n$ range over the positive integers. H conjecture. The number $n^2+1$ is prime for infinitely many values of $n$. This is a classical special case of the broader problem of de…

Let $f$ be a polynomial representing infinitely many distinct primes, let $g\in G$, and let $c_g(f)$ denote the number of primes represented by $f$ for which $g$ is a primitive roo…

Let $P_{\rm MCheb}(z):=z^2-1.87450961730020085\ldots$ be the real quadratic polynomial constructed so that its first-return combinatorics imitate those of the Chebyshev polynomial…

Quadratic Siegel-disk radius conjecture. The correct order is conjectured to be equal to $2$.

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…

Let $X_0\in\mathbb N$ and $\varrho>0$, and let $X$ be sufficiently large in terms of $X_0$ and $\varrho$. Suppose that $A,B\subseteq [X]$ and that … for all primes…

Let $\mathcal A$ be a set of positive integers. For each prime $p$, let $\mathcal A_p$ denote the set of residue classes occupied by $\mathcal A$ modulo $p$. A rational quadratic i…

Doyle's 46-portrait conjecture. The set $\overline{\Gamma}(2)$ consists of precisely $46$ portraits.

Quadratic portrait classification conjecture. Fix an integer $d\geq 1$. There exists a minimal finite set $\overline{\Gamma}=\overline{\Gamma}(d)$ of portraits such that for every…

Let $(A_n)$ be a sequence of integers. An integer $d>1$ is a primitive divisor of $A_n$ if $d\mid A_n$ and $(d,A_m)=1$ for every nonzero term $A_m$ with $m<n$. For $b\in\mathbb{Z}$…