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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.
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 \(k(N)\) 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 \(\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}\)?
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…