Open-problem statements, with their sources attached.

Does there exists an entire non-zero transcendental function f : ℂ → ℂ such that for any sequence n₀ < n₁ < ..., { z | ∃ k, iteratedDeriv (n k) f z = 0 } is dense.

Is $e ^ e$ irrational?

De Giorgi's conjecture holds in dimension $n ≤ 8$.

The density of hyperbolicity conjecture for Multibrot sets, stating that the set of all parameters c for which fun z ↦ z ^ n + c has an attracting cycle is dense in multibrotSet n. Note that we need to require 2 ≤ n because the conjecture is trivially false for n = 1.

Fuglede's conjecture* in one dimension: A bounded subset of ℝ with positive Lebesgue measure is spectral iff it tiles ℝ by translation.

If $A\subseteq\{1, ..., N\}$ with $|A| = n$ is such that the subset sums $\sum_{a\in S}a$ are distinct for all $S\subseteq A$ then

Estimate a lower bound for$\alpha(n)$.

Erdős conjectured that the triangular lattice is best possible in 2D, in particular that $f_2(3n^2 + 3n + 1) < 9n^2 + 3n$.

Note: in [Er75f] is read $9n^2 + 6n$, but this seems to be a typo.

Problem 14 in [Ar2013]: Is it possible to represent an arbitrary compact hausdorff space as an image of a homogeneous compact space under a continuous mapping?

Conjecture 4.3*: For any prime $k$ larger than $3119$, $S(k) > 3k$.

Let $f(z)=\sum_{0\leq k\leq n} \epsilon_k z^k$ be a random polynomial, where $\epsilon_k\in \{0,1\}$ independently uniformly at random for $0\leq k\leq n$.

Is it true that, if $R_n$ is the number of roots of $f(z)$ in $\{ z\in \mathbb{C} : \lvert z\rvert \leq 1\}$, then

almost surely?

Benchmark open subproblem: existence of a SIC-POVM in dimension $68$.

The Köthe conjecture: for any nil ideal I of R, the matrix ideal M_2(I) is a nil ideal of the matrix ring M_2(R).

Let $G$ be a finite $p$-group and assume that all abelian normal subgroups of $G$ have order at most $p^k$. Is it true that every abelian subgroup of $G$ has order at most $p^{2k}$?

Conjecture: Voronovskaja-type formula for Bézier-Bernstein operators with shape parameter $\alpha > 0$, $\alpha \neq 1$.

The source asks for sufficiently smooth functions. This concrete version uses ContDiffOn ℝ 2 f I as a readable baseline regularity assumption; since the domain is the compact interval $[0,1]$, this also explains why no separate boundedness assumption is included here. The variants below record the unknown smoothness threshold more explicitly.

Erdős Problem 70*: Let $\mathfrak{c}$ be the cardinality of the continuum, let $\beta$ be a countable ordinal, and let $2 \le n < \omega$. Is it true that $\mathfrak{c} \to (\beta, n)^3_2$?

Note: The cases $n \le 3$ are trivially true (see omega_three), so the genuine content of the conjecture begins at $n = 4$.

Ahlfors and Grunsky also conjectured in [AG37] that this upper bound is the precise value of the Bloch constant.

Is $\pi ^ e$ irrational?

De Giorgi's conjecture holds in dimension $n = 4$.

The boundary of the Mandelbrot set is conjectured to have zero area.