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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.
Is it true that, for any \(C > 0\), there infinitely many \(n\) such that: \[ p_{n + 1} - p_n > C \frac{\log\log n\log\log\log\log n}{(\log\log\log n) ^ 2}\log n \]
Let \(C\geq 0\). Is there an infinite sequence of \(n_i\) such that \[\lim_{i\to \infty}\frac{p_{n_i+1}-p_{n_i}}{\log n_i}=C?\] We formalise "an infinite sequence of \(n_i\)" as a strictly monotone sequence of indices n : ℕ → ℕ. Note that…
There are infinitely many \(n\) such that \(d_n < d_{n+1} < d_{n+2}\), where \(d\) denotes the prime gap function.
Is the upper density of the set of odd numbers that cannot be expressed as a prime plus two powers of 2 positive?
Is there some \(k\) such that every large integer is the sum of a prime and at most \(k\) powers of \(2\)?
Is it true that \(\sum_{n=1}^\infty(-1)^n\frac{n}{p_n}\) converges, where \(p_n\) is the sequence of primes? Note: In the problem statement, \(p_n\) is the \(n\)-th prime, indexed such that \(p_1=2, p_2=3, \ldots\). We 0-index here to…
Is the set of odd integers not of the form \(2^k+p\) the union of an infinite arithmetic progression and a set of density \(0\)? Erdős called this conjecture "rather silly". Chen [Ch23] has proved the answer is no. This was formalized in…
cluster primes
**Erdős Problem 17.** Are there infinitely many cluster primes?
Let \(A=\{a_1<\cdots<a_t\}\subseteq \{1,\ldots,N\}\) be such that \(\phi(a_1)<\cdots<\phi(a_t)\). The primes are such an example. Are they the largest possible? Can one show that \(\lvert A\rvert<(1+o(1))\pi(N)\) or even…
Let \(k≥3\). Are there \(k\) consecutive primes in arithmetic progression?
Does the longest arithmetic progression of primes in \(\{1,\ldots,N\}\) have length \(o(\log N)\)?
Is there an integer \(m\) with \((m, 6) = 1\) such that none of \(2^k \cdot 3^\ell \cdot m + 1\) are prime, for any \(k, \ell \ge 0\)?
Green-Tao theorem
Are there arbitrarily long arithmetic progressions of primes? Solution: yes. Ref: Green, Ben and Tao, Terence, _The primes contain arbitrarily long arithmetic progressions_
A conjecture by Heath-Brown: The sum of squares of the first \(N\) gaps between consecutive primes behaves like \(N * (log N)^2\).
Is it true that for all c ≥ 0, the density f c of integers for which (p (n + 1) - p n) / log n < c exists and is a continuous function of c?
Let \(f(n)\) count the number of solutions to \(n=p+2^k\) for prime \(p\) and \(k\geq 0\). Show that \(f(n)=o(\log n)\).
Let \(A\subseteq \mathbb{N}\) be a set such that \(\lvert A\cap \{1,\ldots,N\}\rvert \gg \log N\) for all large \(N\). Let \(f(n)\) count the number of solutions to \(n=p+a\) for \(p\) prime and \(a\in A\). Is it true that…
Let c₁, c₂ > 0. Is it true that for any sufficiently large x, there exists more than c₁ * log x many consecutive primes ≤ x such that the difference between any two is > c₂?
Let \(C > 1\). Does the set of integers of the form \(p + \lfloor C^k \rfloor\), for some prime \(p\) and \(k\geq 0\), have density \(>0\)?
Let \(k\geq 3\). Is there a choice of congruence classes \(a_p\pmod{p}\) for every prime \(p\) such that all sufficiently large integers can be written as \(a_p+tp\) for some prime \(p\) and integer \(t\geq k\)?
Let \(A=\{a_1 < \cdots\}\) be an infinite sequence of integers. Let \(f(n)\) count the number of solutions to \[n=\sum_{u\leq i\leq v}a_i.\] Is there such an \(A\) for which \(f(n)\to \infty\) as \(n\to \infty\)? Tao [Ta26] constructed…
Is Erdos375Prop true?
**Erdős Problem 427**: is it true that, for every \(n\) and \(d\), there exists \(k\) such that \[ d \mid p_{n + 1} + \cdots + p_{n + k}, \] where \(p_r\) denotes the \(r\)th prime?
Is there a set \(A\subseteq \mathbb{N}\) such that, for infinitely many \(n\), all of \(n-a\) are prime for all \(a\in A\) with \(0 < a < n\) and \[\liminf\frac{\lvert A\cap [1,x]\rvert}{\pi(x)}>0?\]
inverse Goldbach problem
Are there two infinite sets \(A\) and \(B\) such that \(A+B\) agrees with the primes up to finitely many exceptions? This was disproved by an internal OpenAI model in September 2026 [OAI26a]; see the linked Lean proof.
Is it true that limsup (fun n => (f n - 2 * n.nth Prime : ℕ∞)) atTop = ⊤?
Let \(\operatorname{lcm}(1, \dots, n)\) denote the least common multiple of \(\{1, \dots, n\}\). Let \(p_k\) be the \(k\)-th prime. Is it true that for all \(k \geq 1\),…
Let \(f(u)\) be the largest \(v\) such that no \(m\in (u,v)\) is composed entirely of primes dividing \(uv\). Estimate \(f(u)\). The estimate \(u + 2 \le f(u) \le u^2\) holds for every \(u \ge 2\). The upper bound is attained when \(u\) is…
Let \(s_t(n)\) be the \(t\)-smooth component of \(n\) - that is, the product of all primes \(p\) (with multiplicity) dividing \(n\) such that \(p<t\). Let \(f(n,t)\) count the number of distinct possible values for \(s_t(m)\) for…
Let \(p(n)\) denote the least prime factor of \(n\). Is there a constant \(C>0\) such that \[\sum_{x\leq n\leq x+C\sqrt{x}(\log x)^2}\frac{p(n)}{n}\gg 1\] for all sufficiently large \(x\)?
Is there a function \(f\) with \(f(n)\to\infty\) as \(n\to\infty\) such that, for all large \(n\), there is a composite number \(m\) such that \[ n + f(n) < m < n + p(m) \] Here \(p(m)\) is the least prime factor of \(m\).
Is it true that, for all sufficiently large \(n\), there exists some \(k\) such that \[ p(n+k)>k^2+1, \] where \(p(m)\) denotes the least prime factor of \(m\)?
**Erdős problem 681.** Is it true that for all large \(n\) there exists \(k\) such that \(n + k\) is composite and \(p(n+k) > k^2\), where \(p(m)\) is the least prime factor of \(m\) ?
Is it true that for almost all \(n\) there exists some \(m\in (p_n,p_{n+1})\) such that \[p(m) \geq p_{n+1}-p_n,\] where \(p(m)\) denotes the least prime factor of \(m\)? This was solved in the affirmative by Gafni and Tao [GaTa25]. This…
Let \(P(n, k)\) be the largest prime factor of \(\binom{n}{k}\). There exists \(c > 0\) such that \(P(n, k) \ge \min(n - k + 1, k^{1 + c})\) for all \(0 < k \le n/2\). Erdős stated this for \(1 \le k \le n\) with the bound…
Let \(f(n)\) be the smallest \(0 \leq k \leq n\) for which the factor of \(\binom{n}{k}\) supported on primes at most \(k\) exceeds \(n^2\). Give bounds for \(f(n)\). An internal OpenAI model [APSSV26] has given an elementary argument…
Let \(\epsilon > 0\) and let \(n\) be large depending on \(\epsilon\). Is it true that for all \(n^{\epsilon} < k \leq n^{1-\epsilon}\) the number of distinct prime divisors of \(\binom{n}{k}\) is \[(1+o(1))\, k \sum_{k < p < n}…
Erdős-Woods conjecture
Can there exist two distinct integers \(x\) and \(y\) such that \(x,y\) have the same prime factors, \(x+1,y+1\) have the same prime factors, and \(x+2,y+2\) also have the same prime factors?
Let \(d_n = p_{n+1} - p_n\), where \(p_n\) is the \(n\)th prime. Let \(h(x)\) be maximal such that for some \(n < x\) the numbers \(d_n, d_{n+1}, \ldots, d_{n+h(x)-1}\) are all distinct [Er85c]. Estimate \(h(x)\). In particular, is it true…
Let \(d_n = p_{n+1} - p_n\), where \(p_n\) is the \(n\)th prime. Let \(r(x)\) be the smallest even integer \(t\) such that \(d_n = t\) has no solutions for \(n \le x\). Is it true that \(r(x) \to \infty\)?
second Hardy-Littlewood conjecture
Erdős Problem 855 (Segal's conjecture): \(\pi(x + y) \le \pi(x) + \pi(y)\) for all sufficiently large \(x, y\), i.e. for all \(x, y \ge N\) for some \(N\).
Let \(h(n)\) be such that, for any \(m\geq 1\), in the interval \((m,m+h(n))\) there exist distinct integers \(a_i\) for \(1\leq i\leq \pi(n)\) such that \(p_i\mid a_i\), where \(p_i\) denotes the \(i\)th prime. Estimate \(h(n)\).
If \(\omega_k(n)\) counts the number of distinct prime factors of \(n\) which are \(>k\), then is it true that, for every \(k\geq 1\), \[\liminf_{n\to \infty}\sum_{0\leq i < k}\omega_k(n+i)\leq k?\]
Is it true that \(\liminf f(n)=1\)?
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].
A prime \(p\) is in class \(1\) if the only prime divisors of \(p+1\) are \(2\) or \(3\). In general, a prime \(p\) is in class \(r\) if every prime factor of \(p+1\) is in some class \(\leq r-1\), with equality for at least one prime…
Are there infinitely many primes \(p\) such that \(p - k!\) is composite for each \(k\) such that \(1 ≤ k! < p\)?
Let \(d_n=p_{n+1}-p_n\), where \(p_n\) denotes the \(n\)th prime. Is it true that \[\frac{\max_{n < x}d_{n}d_{n-1}}{(\max_{n < x}d_n)^2}\to 0\] as \(x\to \infty\)?
**Erdős Problem 1138.** Let \(x/2 < y < x\) and \(C > 1\). If \(d = \max_{p_n < x}(p_{n+1} - p_n)\), where \(p_n\) denotes the \(n\)-th prime, then is it true that \[\pi(y + Cd) - \pi(y) \sim \frac{Cd}{\log y}\]?
Let \(1\leq u_1 < u_2 < \cdots\) be the sequence of integers with at most \(2\) prime factors. Is it true that \[\limsup_{k \to \infty} \frac{u_{k+1}-u_k}{\log k}=\infty?\]
Are there infinitely many \(n\) such that \(n-k^2\) is prime for all \(k\) with \((n,k)=1\) and \(k^2 < n\)? In [Va99] it is asked whether \(968\) is the largest integer with this property, but this is an error, since for example…
Are there infinitely many \(n > 2\) such that \(n - 2^k\) is prime for all \(k \geq 1\) with \(2^k < n\)? The only known such \(n\) are \(4, 7, 15, 21, 45, 75, 105\) (OEIS [A039669](https://oeis.org/A039669)).
Let \(k\geq 3\). Is it true that, in any finite colouring of the integers, there are monochromatic arithmetic progressions of primes of length \(k\)? It follows from the theorem of Green and Tao [GrTa08] (that any set of primes with…
Is it true that for every \(\epsilon,\eta>0\) there exists a \(k\) such that the density of \(n\) for which \(P(n(n+1)\cdots(n+k))>n^{1-\epsilon}\) is at least \(1-\eta\) (where \(P(m)\) is the greatest prime divisor of \(m\))?
Let \(\epsilon > 0\). Does there exist \(k\) such that, for every \(n\), any primes \(p_1 < \cdots < p_k < n^{1-\epsilon}\) and any sets \(A_i\) of \(\frac{p_i - 1}{2}\) residue classes modulo \(p_i\), the number of \(m \le n\) that avoid…
Let \(G\) be the graph with vertex set those pairs \((x,y)\in \mathbb{N}^2\) with \(\mathrm{gcd}(x,y)=1\), in which we join two vertices if the differ in only one coordinate, and there by \(\pm 1\). Is there a path going to infinity on…