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.
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.
practical numbers
**Conjecture 1.** Are there infinitely many practical numbers \(m\) such that \(h(m) < (\log \log m)^{O(1)}\)? More precisely: does there exist a constant \(C > 0\) such that for infinitely many practical numbers \(m\), we have…
Let \(A\subset\mathbb{N}\) be infinite such that \(\sum_{a \in A} \frac{1}{a} = \infty\). Must there exist some \(k\geq 1\) such that almost all integers have a divisor of the form \(a+k\) for some \(a\in A\)? This was formalized in Lean…
Are there \(n\) such that there is a covering system with moduli the divisors of \(n\) which is 'as disjoint as possible'? That is, for all \(d\mid n\) with \(d>1\) there is an associated \(a_d\) such that every integer is congruent to…
Let \(A\subseteq\mathbb{N}\) be infinite and \(d_A(n)\) count the number of \(a\in A\) which divide \(n\). Is it true that, for every \(k\), \[\limsup_{x\to \infty} \frac{\max_{n<x}d_A(n)}{\left(\sum_{n\in A\cap[1,x)}\frac{1}{n}\right)^k}…
Let \(\tau(n)\) count the divisors of \(n\) and \(\tau^+(n)\) count the number of \(k\) such that \(n\) has a divisor in \([2^k, 2^{k+1})\). Is it true that, for all \(\epsilon > 0\), \[ \tau^+(n) < \epsilon \cdot \tau(n) \] for almost all…
How large must \(y=y(\epsilon,n)\) be such that the number of integers in \((x,x+y)\) with a divisor in \((n,2n)\) is at most \(\epsilon y\)? The bound is required for every \(x\) and every window length at least \(y\), and…
Let \(A\) be the set of all \(n\) such that \(n = d_1 + ⋯ + d_k\) with \(d_i\) distinct proper divisors of \(n\), but this is not true for any \(m ∣ n\) with \(m < n\). Does: \[ \sum_{n ∈ A} \frac 1 n \] converge? Yes: the sum converges.…
Are there any odd weird numbers?
Let \(1=d_1<\cdots <d_{\tau(n)}=n\) be the divisors of \(n\) and \[G(n) = \sum_{1\leq i<\tau(n)}\frac{d_i}{d_{i+1}}.\] Is it true that \(G(n)\to \infty\) for almost all \(n\)? The answer is yes: Tao observed that…
Let \(\delta_1(n,m)\) be the density of the set of integers with exactly one divisor in \((n,m)\). Is \(\delta_1(n,m)\) unimodular for \(m>n+1\) (i.e. increases until some \(m\) then decreases thereafter)? Cambie has calculated that…
Is it true that \(H(n)/h(n)\to \infty\) for almost all \(n\)? Formalised as: for every threshold \(M\), the set of \(n\) with \(h(n) > 0\) and \(H(n)/h(n) > M\) has natural density \(1\). This is false: in fact, \(H(n)/h(n) = 2 + o(1)\)…
For each \(m\) and \(\alpha\), the density of the set of integers which are divisible by some \(d \equiv 1 \pmod{m}\) with \(1 < d < \exp (m ^ \alpha)\) exists.
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…
The density of the divisor sum set is asymptotically equivalent to \(c_1 / \log(t)^{c_2}\).
For a natural number n, let \(1 = d_1 < \dotsc < d_{\tau(n)} = n\) denote the divisors of \(n\) in increasing order. Does it hold that…
Is it true that, for every \(k \geq 1\), there exist integers \(N_1 < \dots < N_k\) such that \(|\cap_i D(N_i)| \geq k\)?
Let \(\epsilon>0\). Is it true that, for all large \(n\), the number of divisors of \(n\) in \((n^{1/2},n^{1/2}+n^{1/2-\epsilon})\) is \(O_\epsilon(1)\)? Erdős attributes this conjecture to Ruzsa.
Is there an absolute constant \(K\) such that, for every \(C > 0\), if \(n\) is sufficiently large then \(n\) has at most \(K\) divisors in \((n^{\frac{1}{2}}, n^{\frac{1}{2}} + C n^{\frac{1}{4}})\).
Does the limit \(\lim_{n\to\infty} \frac{f(2n)}{f(n)}\) tend to infinity? (Other finite limits have been ruled out by [KoLu25], see below)
Is it true that \(F(x) \leq (\log x)^{O(1)}\)?
There are infinitely many \(n\) such that \(τ(n) = τ(n+1)\). Proved in [He84]. Here τ is the divisor counting function, which is σ 0 in mathlib.
For an irreducible polynomial \(f \in \mathbb{Z}[x]\) with \(f(n) \ge 1\) for sufficiently large \(n\), does there exists a constant \(c = c(f) > 0\) such that \(\sum_{n \le x} \tau(f(n)) \approx c \cdot x \log x\)? Note that it is unclear…
Let \(f(n)\) be the minimal integer \(m\) such that \(n\) is the sum of the \(k\) smallest divisors of \(m\) for some \(k\geq 1\). Is it true that \(f(n)=o(n)\)? The strong claim that \(f(n)=o(n)\) was disproved by Tao in the comments to…
Let \(1=d_1<\cdots<d_{\tau(n)}=n\) be the divisors of \(n\), and for \(\alpha>1\) let \[h_\alpha(n) = \sum_i \left( \frac{d_{i+1}}{d_i}-1\right)^\alpha.\] Is it true that \[\liminf_{n\to \infty}h_\alpha(n) \ll_\alpha 1?\] Erdős [Er81h]…