Regenerated from the community database (Apache-2.0), dated, and linked back to it from every problem. erdosproblems.com no longer shows statuses (Bloom's reasons); this is the database's record, not our verdict.
963 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.
Let \(k\geq 2\). Is there an integer \(n_k\) such that, if \(D=\{ 1<d<n_k : d\mid n_k\}\), then for any \(k\)-colouring of \(D\) there is a monochromatic subset \(D'\subseteq D\) such that \(\sum_{d\in D'}\frac{1}{d}=1\)? This follows from…
Does every finite colouring of the integers have a monochromatic solution to \(1=\sum \frac{1}{n_i}\) with \(2\leq n_1<\cdots <n_k\)? The answer is yes, as proved by Croot [Cr03] - indeed, there are infinitely many disjoint such…
If \(\delta>0\) and \(N\) is sufficiently large in terms of \(\delta\), and \(A\subseteq\{1,\ldots,N\}\) is such that \(\sum_{a\in A}\frac{1}{a}>\delta \log N\) then must there exist \(S\subseteq A\) such that…
Let \(F(k)\) be the number of solutions to \[ 1= \frac{1}{n_1}+\cdots+\frac{1}{n_k},\] where \(1\leq n_1<\cdots<n_k\) are distinct integers. Find good estimates for \(F(k)\).
Let \(x>0\) be a real number. For any \(n\geq 1\) let \[R_n(x) = \sum_{i=1}^n\frac{1}{m_i}<x\] be the maximal sum of \(n\) distinct unit fractions which is \(<x\). Is it true that, for almost all \(x\), for sufficiently large \(n\), we…
Erdős-Straus conjecture
For every \(n>2\) there exist distinct integers \(1 ≤ x < y < z\) such that \(\frac 4 n = \frac 1 x + \frac 1 y + \frac 1 z\).
Let \(A\subseteq \mathbb{N}\) be an infinite set and consider the following greedy algorithm for a rational \(x\in (0,1)\): choose the minimal \(n\in A\) not used so far such that \(n\geq 1/x\) and repeat with \(x\) replaced by…
Let \(p\colon \mathbb{Z} \rightarrow \mathbb{Z}\) be a polynomial (with rational coefficients, taking integer values at all integers) whose leading coefficient is positive and such that there exists no \(d≥2\) with \(d ∣ p(n)\) for all…
Let \(f(k)\) be the minimal value of \(n_k\) such that there exist \(n_1 < n_2 < \dots < n_k\) with \[ 1 = \frac{1}{n_1} + \cdots + \frac{1}{n_k}. \] Is it true that \[ f(k) = (1 + o(1)) \frac{e}{e - 1} k ? \] Proved by Martin [Ma00].…
Let \(k\geq 2\). Is it true that there exists an interval \(I\) of width \((e-1+o(1))k\) and integers \(n_1<\cdots<n_k\in I\) such that \[1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}?\] The answer is yes, proved by Croot [Cr01].
Let \(k\geq2\). Is it true that, for any distinct integers \(1 < n_1 < \cdots < n_k\) such that \(\sum_{i=1}^k \frac{1}{n_i} = 1\), we must have \(\max(n_{i+1} - n_i) \geq 3\)?
Is it true that there are only finitely many pairs of intervals \(I_1\), \(I_2\) such that \[ \sum_{n_1 \in I_1} \frac{1}{n_1} + \sum_{n_2 \in I_2} \frac{1}{n_2} \in \mathbb{N}? \]
Is it true that, for all sufficiently large \(k\), there exist finite intervals \(I_1, \dotsc, I_k \subset \mathbb{N}\), distinct, not overlapping or adjacent, with \(|I_i| \geq 2\) for \(1 \leq i \leq k\) such that \[ 1 = \sum_{i=1}^k…
Let \(a\geq 1\). Must there exist some \(b>a\) such that \[\sum_{a\leq n\leq b}\frac{1}{n}=\frac{r_1}{s_1}\textrm{ and } \sum_{a\leq n\leq b+1}\frac{1}{n}=\frac{r_2}{s_2},\] with \((r_i,s_i)=1\) and \(s_2<s_1\)? If so, how does this…
Let \(n\geq 1\) and define \(L_n\) to be the least common multiple of \(\{1,\ldots,n\}\) and \(a_n\) by \(\sum_{1\leq k\leq n}\frac{1}{k}=\frac{a_n}{L_n}\). Is it true that \((a_n,L_n)=1\) occurs for infinitely many \(n\)?
Let \(A\) be the set of \(n\in \mathbb{N}\) such that there exist \(1\leq m_1<\cdots <m_k=n\) with \(\sum\tfrac{1}{m_i}=1\). Explore \(A\). In particular, does \(A\) have density \(1\)? Straus observed that \(A\) is closed under…
ambiguous statement
Let \(N\geq 1\) and let \(t(N)\) be the least integer \(t\) such that there is no solution to \[1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}\] with \(t=n_1<\cdots <n_k\leq N\). Estimate \(t(N)\). Solved by Liu and Sawhney [LiSa24, Theorem 1.6] (up…
Let \(k(N)\) denote the smallest \(k\) such that there exists \(N ≤ n_1 < ⋯ < n_k\) with \(\frac 1 {n_1} + ... + \frac 1 {n_k} = 1\) Is it true that \(\lim_{N \to \infty} k(N) - (e - 1)N = \infty\)?
Let \(N\geq 1\) and let \(k(N)\) be maximal such that there are \(k\) disjoint \(A_1,\ldots,A_k\subseteq \{1,\ldots,N\}\) with \(\sum_{n\in A_i}\frac{1}{n}=1\) for all \(i\). Estimate \(k(N)\). Is it true that \(k(N)=o(\log N)\)? Hunter…
Does every set \(A \subseteq \mathbb{N}\) of positive density contain some finite \(S \subset A\) such that \(\sum_{n \in S} \frac{1}{n} = 1\)? The answer is yes, proved by Bloom [Bl21] (even if 'positive density' is interpreted as…
Is there an infinite sequence \(a_1 < a_2 < \dots\) such that \(a_{i+1} - a_i = O(1)\) and no finite sum of \(\frac{1}{a_i}\) is equal to 1? There does not exist such a sequence, which follows from the positive solution to…
Let \(A(N)\) denote the maximal cardinality of \(A\subseteq \{1,\ldots,N\}\) such that \(\sum_{n\in S}\frac{1}{n}\neq 1\) for all \(S\subseteq A\). Estimate \(A(N)\). Erdős and Graham [ErGr80] believe the answer is \(A(N)=(1+o(1))N\).…
Let \(f(N)\) be the size of the largest \(A\subseteq \{1,\ldots,N\}\) such that there are no solutions to \[\frac{1}{a}= \frac{1}{b}+\frac{1}{c}\] with distinct \(a,b,c\in A\)? Estimate \(f(N)\). The colouring version of this is [303],…
Is it true that in any finite colouring of the integers there exists a monochromatic solution to \(\frac 1 a = \frac 1 b + \frac 1 c\) with distinct \(a, b, c\)? This is true, as proved by Brown and Rödl [BrRo91]. This was formalized in…
Is it true that \[N(b) \ll \log \log b\]? Yes. This was proved by an internal OpenAI model in September 2026 [OAI26a]; see the linked Lean proof.
For integers \(1\leq a<b\) let \(D(a,b)\) be the minimal value of \(n_k\) such that there exist integers \(1\leq n_1<\cdots <n_k\) with \[\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.\] Estimate \(D(b)=\max_{1\leq a<b}D(a,b)\). Is it…
Let \(\frac a b\in \mathbb{Q}_{>0}\) with \(b\) squarefree. Are there integers \(1 < n_1 < \dots < n_k\), each the product of two distinct primes, such that \(\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}\)? The answer is yes [Li26].
Are there two finite set of primes \(P\) and \(Q\) such that \[ 1 = \left( \sum_{p \in P} \frac{1}{p} \right) \left( \sum_{q \in Q} \frac{1}{q} \right) \] ? Asked by Barbeau [Ba76]. [Ba76] Barbeau, E. J., _Computer challenge corner:…
Let \(N\geq 1\). What is the smallest integer not representable as the sum of distinct unit fractions with denominators from \(\{1,\ldots,N\}\)? This was essentially solved by Croot [Cr99] (see erdos_308.variants.croot); in particular, for…
Let \(N\geq 1\). How many integers can be written as the sum of distinct unit fractions with denominators from \(\{1,\ldots,N\}\)? Are there \(o(\log N)\) such integers? If the number of such integers is \(N(n)\) then it is trivial that…
Let \(\alpha >0\) and \(N\geq 1\). Is it true that for any \(A\subseteq \{1,\ldots,N\}\) with \(\lvert A\rvert \geq \alpha N\) there exists some \(S\subseteq A\) such that \[\frac{a}{b}=\sum_{n\in S}\frac{1}{n}\] with…
Let \(\delta(N)\) be the minimal non-zero value of \(\lvert 1-\sum_{n\in A}\frac{1}{n}\rvert\) as \(A\) ranges over all subsets of \(\{1,\ldots,N\}\). Is it true that \[\delta(N)=e^{-(c+o(1))N}\] for some constant \(c\in (0,1)\)? The…
Does there exist a constant c > 0 such that, for any K > 1, whenever A is a sufficiently large finite multiset of integers with \(\sum_{n \in A} 1/n > K\) there exists some \(S \subseteq A\) such that…
Are there infinitely many pairs (m, P) where m ≥ 2 is an integer and P is a set of distinct primes such that the following equation holds: \(\sum_{p \in P} \frac{1}{p} = 1 - \frac{1}{m}\)?
Let \(n\geq 1\) and let \(m\) be minimal such that \(\sum_{n\leq k\leq m}\frac{1}{k}\geq 1\). We define \[\epsilon(n) = \sum_{n\leq k\leq m}\frac{1}{k}-1.\] How small can \(\epsilon(n)\) be? Is it true that \[\liminf n^2\epsilon(n)=0?\]…
Let \(u_1=1\) and \(u_{n+1}=u_n(u_n+1)\), so that \(\sum_{k\geq 1}\frac{1}{u_k+1}\) and \(u_k=\lfloor c_0^{2^k}+1\rfloor\) for \(k\geq 1\), where \[c_0=\lim u_n^{1/2^n}=1.264085\cdots.\] Let \(a_1<a_2<\cdots \) be any other sequence with…
Is it true that if \(A \subseteq \mathbb{N}\setminus\{1\}\) is a finite set with \(\sum_{n \in A} \frac{1}{n} < 2\) then there is a partition \(A=A_1 \sqcup A_2\) such that \(\sum_{n \in A_i} \frac{1}{n} < 1\) for \(i=1,2\)? This is not…
Is there some constant \(c>0\) such that for every \(n\geq 1\) there exists some \(\delta_k\in \{-1,0,1\}\) for \(1\leq k\leq n\) with \[0< \left\lvert \sum_{1\leq k\leq n}\frac{\delta_k}{k}\right\rvert < \frac{c}{2^n}?\]
There exists a set A with positive density that does not have property P₁. #TODO: prove this lemma by assuming erdos_318.contain_single_even. The density sits in an existential, so HasPosDensity is the *stronger* reading here and weakening…
What is the size of the largest \(A\subseteq\{1, \dots, N\}\) such that there is a function \(\delta : A \to \{-1, 1\}\) such that \[ \sum_{n\in A} \frac{\delta n}{n} = 0 \] and \[ \sum_{n\in A'}\frac{\delta n}{n} \neq 0 \] for all…
Let \(R(N)\) be the size of the largest \(A\subseteq\{1, ..., N\}\) such that all sums \(\sum_{n\in S} \frac{1}{n}\) are distinct for \(S\subseteq A\). What is \(R(N)\)?
Is there a lacunary sequence \(A\subseteq \mathbb{N}\) (so that \(A=\{a_1 < \cdots\}\) and there exists some \(\lambda > 1\) such that \(a_{n+1}/a_n\geq \lambda\) for all \(n\geq 1\)) such that \[\left\{ \sum_{a\in A'}\frac{1}{a} :…
Is there an absolute constant \(C > 0\) such that every integer \(n\) with \(\sigma(n) > Cn\) is the distinct sum of proper divisors of \(n\)? This has been solved in the affirmative by Larsen - in fact, for any \(\epsilon>0\) there exists…