Skip to content

Erdős Problems

1,217 source-owned questions · 604 with a formal statement · searchable by statement, number, topic and source status.

Add a contributionCollection details
Collection coverage

Collection coverage

Source status, exact formal material, and reviewed Results are separate signals.

  • Open per source608
  • Resolved per source556
  • Other source status53
Exact formal statement available604 / 1,217
With Repository-reviewed evidence2 / 1,217
More filters

Coverage is source-observation coverage, not Problem completeness. Inspect coverage

Problems

61 Problems

1/2

NumberQuestionOpen
#4Is it true that, for any C>0C > 0, there infinitely many nn such that: pn+1−pn>Clog⁡log⁡nlog⁡log⁡log⁡log⁡n(log⁡log⁡log⁡n)2log⁡n p_{n + 1} - p_n > C \frac{\log\log n\log\log\log\log n}{(\log\log\log n) ^ 2}\log n provedFormalized
#5No statement retained — open to read what the source holdsopenNo formal declaration
#6There are infinitely many nn such that dn<dn+1<dn+2d_n < d_{n+1} < d_{n+2}, where dd denotes the prime gap function.provedFormalized
#9Is the upper density of the set of odd numbers that cannot be expressed as a prime plus two powers of 2 positive?openFormalized
#10Is there some kk such that every integer is the sum of a prime and at most kk powers of 22?openFormalized
#15Is it true that ∑n=1∞(−1)nnpn\sum_{n=1}^\infty(-1)^n\frac{n}{p_n} converges, where pnp_n is the sequence of primes?openFormalized
#16Is the set of odd integers not of the form 2k+p2^k+p the union of an infinite arithmetic progression and a set of density 00?disproved (Lean)Formalized
#17Erdős Problem 17. Are there infinitely many cluster primes?openFormalized
#49No statement retained — open to read what the source holdsprovedNo formal declaration
#141Let k≥3k≥3. Are there kk consecutive primes in arithmetic progression?openFormalized
#200Does the longest arithmetic progression of primes in {1,…,N}\{1,\ldots,N\} have length o(log⁡N)o(\log N)?openFormalized
#203Is there an integer mm with (m,6)=1(m, 6) = 1 such that none of 2k⋅3ℓ⋅m+12^k \cdot 3^\ell \cdot m + 1 are prime, for any k,ℓ≥0k, \ell \ge 0?openFormalized
#218The set of indices nn for which a prime gap is preceded by a larger or equal prime gap has a natural density of 12\frac 1 2.openFormalized
#219Are there arbitrarily long arithmetic progressions of primes? Solution: yes. Ref: Green, Ben and Tao, Terence, _The primes contain arbitrarily long arithmetic progressions_provedFormalized
#233A conjecture by Heath-Brown: The sum of squares of the first NN gaps between consecutive primes behaves like N∗(logN)2N * (log N)^2.openFormalized
#234Is 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?openFormalized
#236Let f(n)f(n) count the number of solutions to n=p+2kn=p+2^k for prime pp and k≥0k\geq 0. Show that f(n)=o(log⁡n)f(n)=o(\log n).openFormalized
#237No statement retained — open to read what the source holdsproved (Lean)No formal declaration
#238Let 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?openFormalized
#240No statement retained — open to read what the source holdsprovedNo formal declaration
#244Let C>1C > 1. Does the set of integers of the form p+⌊Ck⌋p + \lfloor C^k \rfloor, for some prime pp and k≥0k\geq 0, have density >0>0?openFormalized
#279Let k≥3k\geq 3. Is there a choice of congruence classes ap(modp)a_p\pmod{p} for every prime pp such that all sufficiently large integers can be written as ap+tpa_p+tp for some prime pp and integer t≥kt\geq k?openFormalized
#358When An=nA_n = n, the function ff defined above counts the number of odd divisors of nn.provedFormalized
#427Erdős Problem 427: is it true that, for every nn and dd, there exists kk such that d∣pn+1+⋯+pn+k, d \mid p_{n + 1} + \cdots + p_{n + k}, where prp_r denotes the rrth prime?proved (Lean)Formalized
#428Is there a set A⊆NA\subseteq \mathbb{N} such that, for infinitely many nn, all of n−an-a are prime for all a∈Aa\in A with 0<a<n0 < a < n and lim inf⁡∣A∩[1,x]∣π(x)>0?\liminf\frac{\lvert A\cap [1,x]\rvert}{\pi(x)}>0?openFormalized
#431No statement retained — open to read what the source holdsopenNo formal declaration
#454Is it true that limsup (fun n => (f n - 2 * n.nth Prime : ℕ∞)) atTop = ⊤?openFormalized
#458Let lcm⁡(1,…,n)\operatorname{lcm}(1, \dots, n) denote the least common multiple of {1,…,n}\{1, \dots, n\}. Let pkp_k be the kk-th prime. Is it true that for all k≥1k \geq 1, lcm⁡(1,…,pk+1−1)<pk⋅lcm⁡(1,…,pk)\operatorname{lcm}(1, \dots, p_{k+1}-1) < p_k \cdot \operatorname{lcm}(1, \dots, p_k)?falsifiableFormalized
#459Let f(u)f(u) be the largest vv such that no m∈(u,v)m\in (u,v) is composed entirely of primes dividing uvuv. Estimate f(u)f(u).solved (Lean)Formalized
#461No statement retained — open to read what the source holdsopenNo formal declaration
#462No statement retained — open to read what the source holdsopenNo formal declaration
#463Is there a function ff with f(n)→∞f(n)\to\infty as n→∞n\to\infty such that, for all large nn, there is a composite number mm such that n+f(n)<m<n+p(m) n + f(n) < m < n + p(m) Here p(m)p(m) is the least prime factor of mm.openFormalized
#680Is it true that, for all sufficiently large nn, there exists some kk such that p(n+k)>k2+1, p(n+k)>k^2+1, where p(m)p(m) denotes the least prime factor of mm?openFormalized
#681Erdős problem 681. Is it true that for all large nn there exists kk such that n+kn + k is composite and p(n+k)>k2p(n+k) > k^2, where p(m)p(m) is the least prime factor of mm ?openFormalized
#682No statement retained — open to read what the source holdsprovedNo formal declaration
#683There exists c>0c > 0 such that P(n,k)>min⁡{n−k+1,k1+c}P(n, k) > \min\{n-k+1, k^{1 + c}\} for all 0<k<n0 < k < n.}openFormalized
#684No statement retained — open to read what the source holdsopenNo formal declaration
#685No statement retained — open to read what the source holdsopenNo formal declaration
#779A Conjecture of Marian Deaconescu, see p.120 in https://doi.org/10.2307/2975810falsifiableFormalized
#850Can there exist two distinct integers xx and yy such that x,yx,y have the same prime factors, x+1,y+1x+1,y+1 have the same prime factors, and x+2,y+2x+2,y+2 also have the same prime factors?openFormalized
#852No statement retained — open to read what the source holdsopenNo formal declaration
#853Let dn=pn+1−pnd_n = p_{n+1} - p_n, where pnp_n is the nnth prime. Let r(x)r(x) be the smallest even integer tt such that dn=td_n = t has no solutions for n≤xn \le x.openFormalized
#855Erdős Problem 855 (Segal's conjecture): π(x+y)≤π(x)+π(y)\pi(x + y) \le \pi(x) + \pi(y) for sufficiently large x,yx, y.openFormalized
#860No statement retained — open to read what the source holdsopenNo formal declaration
#890If ωk(n)\omega_k(n) counts the number of distinct prime factors of nn which are >k>k, then is it true that, for every k≥1k\geq 1, lim inf⁡n→∞∑0≤i<kωk(n+i)≤k?\liminf_{n\to \infty}\sum_{0\leq i < k}\omega_k(n+i)\leq k?openFormalized
#950Is it true that lim inf⁡f(n)=1\liminf f(n)=1?openFormalized
#997Is it true that, for every α\alpha, the sequence {αpn}\{ \alpha p_n\} is not well-distributed, if pnp_n is the sequence of primes?proved (Lean)Formalized
#1055A prime pp is in class 11 if the only prime divisors of p+1p+1 are 22 or 33. In general, a prime pp is in class rr if every prime factor of p+1p+1 is in some class ≤r−1\leq r-1, with equality for at least one prime factor. Are there infinitely many primes in each class?openFormalized

Search problems.science

Find a Problem, Result, source, or page