56 problems

Frid–Puzynina–Zamboni conjecture. If $k$ is an integer such that

Hof–Knill–Simon conjecture. The language of $\mathbf{u}$ contains infinitely many palindromes if and only if $\mathcal{L}(\mathbf{u})$ coincides with the language of a fixed point…

Brlek–Reutenauer conjecture. If $\mathbf u$ is closed under reversal, then

Let an infinite word be aperiodic if it is not ultimately periodic, and let the palindromic length of a factor be the minimum number of palindromes whose product is that factor. Un…

Let $w$ be an infinite word, and let $\operatorname{PPL}_w(n)$ denote the palindromic length of the prefix of $w$ of length $n$. Frid's conjecture. If $w$ is $k$-power-free for som…

Projection richness conjecture. The word $\zeta(\mathbf u)$ is rich.

Let $\mathfrak{F}^{(a,b)}_n$ denote the standard length-$n$ prefix or level-$n$ finite word in the biperiodic Fibonacci construction, and let $\Phi$ be the palindromisation operato…

Let ${\bf x}=(a_i)_{i\geq 0}$ be an infinite word over a finite alphabet $\Sigma$. Write $\rho_{\bf x}(n)$ for the number of distinct length-$n$ factors of ${\bf x}$, and let…

Prime-palindrome conjecture. There exist infinitely many palindromes which are also prime numbers.

Let a word be decomposable into $k$ palindromes, and consider an allowed reduced form obtained from such a decomposition in the free group. The centers of the initial palindromes a…

Let $L$) be a language on $k$ letters satisfying a symmetric order condition and having no connection. Its palindromic complexity is the number of palindromic factors of each lengt…

Fix a base $b\geq 2$ and let $\overleftarrow{n}$ denote the integer obtained by reversing the base-$b$ digits of $n$. A square-free palindrome is a positive integer that is square-…

Let ${\mathcal B}_n$ be the set of binary integers with $n$ binary digits, let $\overleftarrow{a}$ denote the integer obtained by reversing the binary digits of $a$, and let…

Let $b\geq 2$ be an integer. A positive integer is $b$-palindromic if its sequence of base-$b$ digits is a palindrome, and let $\mathscr{P}_b(x)$ denote the set of positive $b$-pal…

Puzynina–Zamboni conjecture. If the palindromic lengths of all factors of $d4ce$ are bounded, then $d4ce$ is ultimately periodic.

Arbitrarily long gaps conjecture. Consecutive positive integers each not a $v$-palindrome in base $b$ can be arbitrarily long.

Squarefree v-palindromes conjecture. There are infinitely many $v$-palindromes $n$ in base $b$ such that both $n$ and $r_b(n)$ are squarefree.

Refined fundamental-period conjecture. The quantity $\omega_0(n)$ is always $1$ or $\omega_b(n)$.

Type-independence conjecture. The type of $m$ with respect to $n_1$ is the same as the type of $m$ with respect to $n_2$.

Fundamental-period conjecture. Either

Let $\mathcal{S}_k$ denote the set of Hertzsprung patterns of length $k$, and let $\Omega(\sigma,\sigma)$ be the autocorrelation polynomial of $\sigma\in\mathcal{S}_k$. For a palin…

PPL-difference automaticity conjecture. The PPL-difference sequence $d_{\mathbf{u}}$ of a $k$-automatic word is always $k$-automatic.

Frid's conjecture. The palindromic length sequence of an infinite word is bounded if and only if the infinite word is ultimately periodic.

Let $u$ be an infinite word that is $p$-power-free for some $p$, meaning that $u$ contains no factor equal to the $p$th power of a nonempty word. Let $PPL_u(n)$ be the minimum numb…

Let $u=u[1]u[2]\cdots$ be an infinite word. For $n\in\mathbb N$, let $PPL_u(n)$ denote the minimum number of concatenated palindromes needed to express the prefix of $u$ of length…