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908 of 1221 statements are formalized in Lean (Formal Conjectures) · 107 carry an Erdős prize · 3 are independent of ZFC and 9 more can't be proved, or can't be disproved, from ZFC alone · “open” counts the computable and Lean-checked ones too.
If \(p(z) \in \mathbb{C}[z]\) is a monic polynomial of degree \(n\), is the length of the curve \(\{z \in \mathbb{C} : \lvert p(z)\rvert = 1\}\) maximised when \(p(z) = z^n - 1\)? We require \(n \geq 1\). For \(n = 0\) the only monic…
If \(p(z)\) is a polynomial of degree \(n\) such that \(\{z : \lvert p(z)\rvert\leq 1\}\) is connected then is it true that \[\max_{\substack{z\in\mathbb{C}\\ \lvert p(z)\rvert\leq 1}} \lvert p'(z)\rvert \leq (\tfrac{1}{2}+o(1))n^2?\]…
Let \(p(z)=\prod_{i=1}^n (z-z_i)\) for \(\lvert z_i\rvert \leq 1\). Is it true that \[\lvert\{ z: \lvert p(z)\rvert <1\}\rvert>n^{-O(1)}\] (or perhaps even \(>(\log n)^{-O(1)}\))? Conjectured by Erdős, Herzog, and Piranian [EHP58]. The…
Is it true that \(\limsup M_n = \infty\)? This is Problem 4.1 in [Ha74] where it is attributed to Erdős. The weaker conjecture that \(\limsup M_n=\infty\) was proved by Wagner [Wa80], who show that there is some \(c>0\) with…
Let \[ f(\theta) = \sum_{0\leq k\leq n}c_k e^{ik\theta}\] be a trigonometric polynomial all of whose roots are real, such that \(\max_{\theta\in [0,2\pi]}\lvert f(\theta)\rvert=1\). Then \[\int_0^{2\pi}\lvert f(\theta)\rvert…
Is there an entire non-linear function \(f\) such that, for all \(x\in\mathbb{R}\), \(x\) is rational if and only if \(f(x)\) is? Barth and Schneider [BaSc70] proved the stronger result for countable dense subsets of \(\mathbb{R}\).
Let \((S_n)_{n \ge 1}\) be a sequence of sets of complex numbers, none of which have a finite limit point. Does there exist an entire transcendental function \(f(z)\) such that, for all \(n \ge 1\), there exists some \(k_n \ge 0\) such…
Let \(P(z)=\sum_{1\leq k\leq n}a_kz^k\) for some \(a_k\in \mathbb{C}\) with \(\lvert a_k\rvert=1\) for \(1\leq k\leq n\). Does there exist a constant \(c>0\) such that, for \(n\geq 2\), we have \[\max_{\lvert z\rvert=1}\lvert P(z)\rvert…
reverse Littlewood-Offord problem
If \(z_1,\ldots,z_n\in \mathbb{C}\) with \(\lvert z_i\rvert=1\) then is it true that the probability that \[\lvert \epsilon_1z_1+\cdots+\epsilon_nz_n\rvert \leq \sqrt{2},\] where \(\epsilon_i\in \{-1,1\}\) uniformly at random, is…
Let \(f(k)\) be the minimum number of terms in \(P(x)^2\), where \(P \in \mathbb{Q}[x]\) ranges over all polynomials with exactly \(k\) non-zero terms. Is it true that \(f(k) \to \infty\) as \(k \to \infty\)? A conjecture of Erdős and…
strong Littlewood-Offord problem
Let \(z_1,\ldots,z_n\in\mathbb{C}\) with \(1\leq \lvert z_i\rvert\) for \(1\leq i\leq n\). Let \(D\) be an arbitrary disc of radius \(1\). Is it true that the number of sums of the shape \[\sum_{i=1}^n\epsilon_iz_i \textrm{ for…
Let \(f(z) ∈ ℂ[z]\) be a monic non-constant polynomial. Can the set \(\{z ∈ ℂ : |f(z)| ≤ 1\}\) be covered by a set of closed discs the sum of whose radii is \(≤ 2\)?
Chowla's cosine problem
**Chowla's cosine problem** If \(A\subset \mathbb{N}\) is a finite set of positive integers of size \(N > 0\) then is there some absolute constant \(c>0\) and \(\theta\) such that \[\sum_{n\in A}\cos(n\theta) < -cN^{1/2}?\]
Littlewood's conjecture
Is it true that, if \(A\subset \mathbb{Z}\) is a finite set of size \(N\), then \[\int_0^1 \left\lvert \sum_{n\in A}e(n\theta)\right\rvert \mathrm{d}\theta \gg \log N,\] where \(e(x)=e^{2\pi ix }\)? Littlewood's conjecture, proved…
Let f be a transcendental entire function. What is the greatest possible value of liminf (fun r : ℝ => ratio r f) atTop?
Let f = ∑ aₖzⁿₖ be an entire function of finite order such that nₖ / k → ∞. Then limsup (fun r => ratio r f) atTop = 1. This is proved in [Fu63].
If f(z) = ∑ aₖzⁿₖ is an entire function (with aₖ ≠ 0 for all k) such that nₖ / k → ∞, is it true that f assumes every value infinitely often?
Let \(z_1,\ldots,z_n\in \mathbb{C}\) with \(z_1=1\). Must there exist an absolute constant \(c>0\) such that \[ \max_{1\leq k\leq n}\left\lvert \sum_{i}z_i^k\right\rvert>c? \] Atkinson proved that \(c=1/6\) suffices.
Let \((\epsilon_k)_{k\geq 0}\) be independently uniformly chosen at random from \(\{-1,1\}\). If \(R_n\) counts the number of real roots of \(f_n(z)=\sum_{0\leq k\leq n}\epsilon_k z^k\) then is it true that, almost surely, \[\lim_{n\to…
Let \(f(z)=\sum_{0\leq k\leq n} \epsilon_k z^k\) be a random polynomial, where \(\epsilon_k\in \{-1,1\}\) 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…
Is it true that all except at most \(o(2^n)\) many degree \(n\) polynomials with \(\pm 1\)-valued coefficients \(f(z)\) have \(\lvert f(z)\rvert <1\) for some \(\lvert z\rvert=1\)? The answer is yes: Littlewood [Li66] conjectured the…
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.
Let \(f:\mathbb{R}\to \mathbb{R}\) be such that \(f(x+h)-f(x)\) is continuous for every \(h>0\). Is it true that \[f=g+h\] for some continuous \(g\) and additive \(h\) (i.e. \(h(x+y)=h(x)+h(y)\))? A conjecture of Erdős from the early…
Let \(1<a_1<\cdots\) be a sequence of integers such that \(\sum\frac{1}{a_i}<\infty\). Is it true that, for every \(t\in \mathbb{R}\), \[1+\sum_{k}\frac{1}{a_k^{1+it}}\neq 0?\] Yip [Yi25] has proved that this is not always true - in fact,…
Does there exist a constant \(C>1\) such that, for every \(n\geq 2\), there exists a sequence \(z_i\in \mathbb{C}\) with \(z_1=1\) and \(\lvert z_i\rvert \geq 1\) for all \(1\leq i\leq n\) with…
Let \(z_1,\ldots,z_n\in \mathbb{C}\) be a sequence such that \(z_1=1\). Suppose that the sequence of \[s_k=\sum_{1\leq i\leq n}z_i^k\] contains infinitely many \((n-1)\)-tuples of consecutive values of \(s_k\) which are all \(0\). Then…
Question 1: Is it true that \(\limsup_{k \to \infty} A_k = \infty\)? Erdős [Er64b] remarks it is "easy to see" that \(\limsup_k \sup_n |\sum_{j \le n} e(k x_j)| = \infty\). Erdős [Er65b] later found a "very easy" proof that…
Let \(f=a_0+\cdots+a_dx^d\in \mathbb{C}[x]\) be a polynomial. Is it true that, if \(f\) has roots \(z_1,\ldots,z_d\) with corresponding arguments \(\theta_1,\ldots,\theta_d\in [0,2\pi]\), then for all intervals \(I\subseteq [0,2\pi]\) \[…
Let \(E\subseteq (0,1)\) be a meaurable subset with Lebesgue measure \(\lambda(E)\). Is it true that, for almost all \(\alpha\), \[\lim_{n\to \infty}\frac{1}{n}\sum_{1\leq k\leq n}1_{\{k\alpha \}\in E}=\lambda(E)\] for all \(E\)? This is a…
Erdős Problem 995: For every lacunary sequence \((n_k)\) of integers and every \(f \in L^2([0,1])\) with \(\int_0^1 f = 0\), is it true that for almost all \(\alpha\), \[\sum_{k < N} f(\{\alpha n_k\}) = o\!\left(N \sqrt{\log\log…
Does there exists a positive constant C such that for all f ∈ L²[0,1] and all lacunary sequences n, if ‖f - fₖ‖₂ = O(1 / log log log k ^ C), then for almost every x, lim ∑ k ∈ Finset.range N, f (n k • x)) / N = ∫ t, f t ∂t?
Is it true that, for every \(\alpha\), the sequence \(\{ \alpha p_n\}\) is not well-distributed, if \(p_n\) is the sequence of primes? The answer is yes, by [APSSV26, Section 4]; a Lean formalisation is available in [Mo26].
For any \(0<\alpha<1\), let \(f(\alpha,n)=\frac{1}{\log n}\sum_{1\leq k\leq n}(\tfrac{1}{2}- \{ \alpha k\})\). Does \(f(\alpha,n)\) have an asymptotic distribution function? In other words, is there a non-decreasing function \(g\) such…
What is the infimum of |{x ∈ ℝ : |f x| < 1}| over all nonconstant monic polynomials f such that all of its roots are real and contained in [-1,1]?
**Erdős–Herzog–Piranian Component Lemma** (Metric Properties of Polynomials, 1958): If \(f\) is a monic degree \(n\) polynomial with all roots in the unit disk, then some connected component of \(\{z \mid |f(z)| < 1\}\) contains at least…
**Erdős Problem 1043**: Let \(f\in \mathbb{C}[x]\) be a monic polynomial. Must there exist a straight line \(\ell\) such that the projection of \[\{ z: \lvert f(z)\rvert\leq 1\}\] onto \(\ell\) has measure at most \(2\)? Pommerenke [Po61]…
Let \(f(z)=\prod_{i=1}^n(z-z_i)\in\mathbb{C}[x]\) where \(\lvert z_i\rvert\leq 1\) for all \(i\). If \(\Lambda(f)\) is the maximum of the lengths of the boundaries of the connected components of \[ \{ z: \lvert f(z)\rvert<1\} \] then…
Let \(f\in \mathbb{C}[x]\) be a monic polynomial and \[E=\{ z: \lvert f(z)\rvert <1\}.\] If \(E\) is connected then is \(E\) contained in a disc of radius \(2\)? A problem of Erdős, Herzog, and Piranian [EHP58], who also ask, if…
Let \(f\in \mathbb{C}[x]\) be a monic polynomial with \(m\) distinct roots, and let \(c>0\) be a constant small enough such that \(\{ z: \lvert f(z)\rvert\leq c\}\) has \(m\) distinct connected components. Must all these components be…
If \(f\in \mathbb{C}[x]\) is a monic polynomial with all roots satisfying \(\lvert z\rvert \leq r\) for some \(0 < r < 2\), then must \[\{ z: \lvert f(z)\rvert <1\}\] have a connected component with diameter \(>2-r\)? A problem of Erdős,…
Let \(\mathfrak{m}\) be an infinite cardinal with \(\aleph_0 < \mathfrak{m} < \mathfrak{c} = 2^{\aleph_0}\). Let \(\{f_\alpha\}\) be a family of entire functions such that, for every \(z_0 \in \mathbb{C}\), there are at most…
Let \(f:\mathbb{R}\to \mathbb{R}\) be such that \[2f(x) \leq f(x+h)+f(x+2h)\] for every \(x\in \mathbb{R}\) and \(h>0\). Must \(f\) be monotonic? A problem of Kemperman [Ke69], who proved it is true if \(f\) is measurable. Erdős [Er81b]…
If \[f(x+y)=f(x)+f(y)\] for almost all \(x,y\in \mathbb{R}\) then there exists a function \(g\) such that \[g(x+y)=g(x)+g(y)\] for all \(x,y\in\mathbb{R}\) such that \(f(x)=g(x)\) for almost all \(x\). Proved independently by de Bruijn…
Let \(C>0\). There exists \(\epsilon>0\) such that if \(n\) is sufficiently large the following holds. For any \(x_1,\ldots,x_n\in [-1,1]\) there exist \(y_1,\ldots,y_n\in [-1,1]\) such that, if \(P\) is a polynomial of degree…
Is there some constant \(c > 0\) such that, for all large enough \(n\) and all polynomials \(P\) of degree \(n\) with coefficients in \(\{-1, 1\}\), \[\max_{|z|=1} |P(z)| > (1 + c) \sqrt{n}?\] No. This was disproved by an internal OpenAI…
Does there exist, for every \(\alpha \in [0, 1]\), a ring in \(\mathbb{R}\) with Hausdorff dimension \(\alpha\)?
Let \(E\subset (0,\infty)\) be a set of positive measure. Is it true that, for almost all \(x>0\), for all sufficiently large (depending on \(x\)) integers \(n\) there exists an integer \(r\geq 1\) such that \(nx\in r\cdot E\)? A problem…
ambiguous statement