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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.
OPENnumber theory · base representations · complete sequences
Let \(3 \le d_1 < d_2 < \dots < d_r\) be integers such that \[\sum_{1 \le i \le r}\frac 1{d_i - 1} \ge 1.\] Can all sufficiently large integers be written as a sum of the shape \(\sum_i c_ia_i\) where \(c_i \in \{0, 1\}\) and \(a_i\) has…
Let \(A\subseteq \mathbb{N}\) be a complete sequence, and define the threshold of completeness \(T(A)\) to be the least integer \(m\) such that all \(n\geq m\) are in \[P(A) = \left\{\sum_{n\in B}n : B\subseteq A\textrm{ finite }\right\}\]…
Is it true that for every lacunary, strongly complete sequence A that is not complete whenever infinitely many terms are removed from it, lim A (n + 1) / A n = (1 + √5) / 2? The answer is no. A counterexample recorded at…
The set of subset sums of a set A ⊆ ℕ. -/ local notation "𝓟" A => subsetSums A /-- Is there a sequence \(A=\{a_1\leq a_2\leq \cdots\}\) of integers with \[\lim \frac{a_{n+1}}{a_n}=2\] such that \[P(A')= \left\{\sum_{n\in B}n : B\subseteq…
For what values of \(0 \leq m < n\) is there a complete sequence \(A = \{a_1 \leq a_2 \leq \cdots\}\) of integers such that 1. \(A\) remains complete after removing any \(m\) elements, but 2. \(A\) is not complete after removing any \(n\)…
For what values of \(t,\alpha \in (0,\infty)\) is the sequence \(\lfloor t\alpha^n\rfloor\) complete (that is, all sufficiently large integers are the sum of distinct integers of the form \(\lfloor t\alpha^n\rfloor\))?
Let \(p(x) \in \mathbb{Q}[x]\) be a non-constant rational polynomial with positive leading coefficient. Is it true that \[A=\{ p(n)+1/n : n \in \mathbb{N}\}\] is strongly complete, in the sense that, for any finite set \(B\),…
Let \(\alpha,\beta\in \mathbb{R}_{>0}\) such that \(\alpha/\beta\) is irrational. Is the multiset \[\{ \lfloor \alpha\rfloor,\lfloor 2\alpha\rfloor,\lfloor 4\alpha\rfloor,\ldots\}\cup \{ \lfloor \beta\rfloor,\lfloor 2\beta\rfloor,\lfloor…
Source: teorth/erdosproblems (its own interactive table), Apache-2.0, pulled 2026-10-21. A status here is what that database records, not an independent verification by us, except where a problem page says so.