30 problems

Let $p$ and $q$ be monic polynomials of degree $2$, let $V$ be a vector space, and let $(b,u)$ be a symplectic pair on $V$. A symplectic $(p,q)$-difference is a symplectic pair sat…

Let $N\geq 4$. A quadratic polynomial is a polynomial $f(x)\in\mathbb{Q}[x]$ of degree $2$, and a rational point of period $N$ is an $x\in\mathbb{Q}$ such that $f^{(N)}(x)=x$ but…

Let $p$ be a prime, and let $I_p$ denote the number of parameters $a$ for which the functional graph associated with the quadratic polynomial $f_a$ is connected. The computations u…

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 $f\in\mathbb{Z}[x]$ 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 $f$ 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…

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…

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 $n$.

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…

Let $\phi(x)=x^2+c$ for some $c\in\mathbb{Q}$, and let $a\in\mathbb{Q}$ be a periodic point for $\phi$, meaning that $\phi^n(a)=a$ for some $n\geq1$; the least such $n$ is the peri…

For $m\geq 2$ and $n\geq 1$, let $P_{m,n}$ denote the multiplier polynomial in the quadratic case $d=2$. A polynomial is called 2-special when it has the special coefficient and di…

Let $f_c(x)=x^2+c\in\mathbb{Q}[x]$ be a quadratic polynomial, and let a rational point have exact period $n$ if its period is $n$ and no smaller positive period. Uniform Boundednes…

Flynn–Poonen conjecture. There is no quadratic polynomial $f(x)\in{\mathbb Q}[x]$ with a rational point of exact period $N$.

Let $S=\{x^2+c_1,x^2+c_2,\dots,x^2+c_s\}$ be a set of distinct quadratic polynomials with $c_i\in\mathbb{Q}$. A point $P\in\mathbb{Q}$ has finite orbit for $S$ if its forward orbit…

Let $W_n$ be the iterated wreath-product group acting on the rooted binary tree of height $n$, let $K_n=\operatorname{Ker}(\pi_n:W_n\twoheadrightarrow W_{n-1})$, and let…

Let $a,b,c$ be positive integers with $2 mid ab$ and $2 divides c$. A practical number is a positive integer each of whose integers from $1$ through it can be written as a sum…

Let $P$ and $Q$ be quadratic polynomials with locally connected Julia sets, and let $p$ and $q$ denote their corresponding parameters in the Mandelbrot set. The quadratic mating co…

Let $A\subset [1,N]$ be a set of integers. For each prime $p$, write $A_p$ for the set of residue classes modulo $p$ represented by elements of $A$. The notation $X\ll Y$ means tha…

Let $p\geq3$ be prime. Let ${\mathscr A}_p$ and ${\mathscr B}_p^*$ be the parameter sets defined in the paper, with ${\mathscr A}_p$ denoting the relevant set of parameters and…

Let $p$ be prime, and let $I_p$ denote the number of parameters $a\in\mathbb F_p$ for which the functional graph ${\mathcal G}_a$ of $f_a(X)=X^2+a$ is connected. The existence conj…

Quadratic-prime conjecture. There are infinitely many primes of the form $p=n^2+1$, and

Let $f\in\mathbf Q[x]$ be a quadratic polynomial, and let $n>3$ be an integer. Let $\Phi_{n,f}$ denote the period-$n$ dynatomic polynomial of $f$, and let $\mathcal S_{\Phi_{n,f}}$…

Let $f\in\mathbf Q[x]$ be a quadratic polynomial, and let $n>3$ be an integer. For a prime $p$, write $\mathbf Q_p$ for the field of $p$-adic numbers. A point has period $n$ for…

Nonexistence conjecture. There are no rational values $c$ such that $\phi_c$ has a 5-periodic point in $\mathbb{Q}_{\mathrm{quad}}$.