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Erdős Problems

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

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576 Problems · 1 with reviewed evidence

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NumberQuestionOpen
#672Can the product of an arithmetic progression of positive integers n,n+d,...,n+(k−1)dn, n + d, ..., n + (k - 1)d of length ≥ 4, with (n,d)=1(n, d) = 1, be a perfect power?verifiableFormalized
#673No statement retained — open to read what the source holdsprovedNo formal declaration
#674Are there any integer solutions to xxyy=zzx^xy^y=z^z with x,y,z>1x,y,z>1?proved (Lean)Formalized
#675No statement retained — open to read what the source holdsopenNo formal declaration
#676No statement retained — open to read what the source holdsopenNo formal declaration
#677Denote by M(n,k)M(n, k) the least common multiple of the finite set {n+1,…,n+k}\{n+1, \dotsc, n+k\}. Is it true that for all m≥n+km \geq n + k, we get M(m,k)≠M(n,k)M(m, k) \neq M(n, k)?openFormalized
#678Write M(n,k)M(n, k) be the least common multiple of {n+1,…,n+k}\{n+1, \dotsc, n+k\}. Let kk be sufficiently large. Are there infinitely many m,nm, n with m≥n+km \geq n + k such that M(n,k)>M(m,k+1) M(n, k) > M(m, k + 1) ? The answer is yes, as proved in a strong form by Cambie [Ca24]. [Ca24] S. Cambie, Resolution of an Erdős' problem on least common multiples. arXiv:2410.09138 (2024).proved (Lean)Formalized
#679No statement retained — open to read what the source holdsopenNo formal declaration
#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
#686Can every integer N≥2N≥2 be written as N=∏1≤i≤k(m+i)∏1≤i≤k(n+i)N=\frac{\prod_{1\leq i\leq k}(m+i)}{\prod_{1\leq i\leq k}(n+i)} for some k≥2k≥2 and m≥n+km≥n+k?openFormalized
#687No statement retained — open to read what the source holdsopenNo formal declaration
#688In particular, is it true that ϵn=o(1)\epsilon_n = o(1)?openFormalized
#689Let n be sufficiently large. Is there some choice of congruence class a_p for all primes 2 ≤ p ≤ n such that every integer in [1,n] satisfies at least two of the congruences ≡ a_p (mod p)?openFormalized
#690No statement retained — open to read what the source holdssolvedNo formal declaration
#691No statement retained — open to read what the source holdsopenNo formal declaration
#692Let δ1(n,m)\delta_1(n,m) be the density of the set of integers with exactly one divisor in (n,m)(n,m). Is δ1(n,m)\delta_1(n,m) unimodular for m>n+1m>n+1 (i.e. increases until some mm then decreases thereafter)?disproved (Lean)Formalized
#693No statement retained — open to read what the source holdsopenNo formal declaration
#694Let f_\max(n) be the largest mm such that ϕ(m)=n\phi(m) = n, and f_\min(n) be the smallest such mm, where ϕ\phi is Euler's totient function. Investigate \max_{n\leq x}\frac{f_\max(n)}{f_\min(n)}. solved (Lean)Formalized
#695Let q1<q2<⋯q_1 < q_2 < \cdots be a sequence of primes such that qi+1≡1(modqi)q_{i + 1} \equiv 1 \pmod{q_i}. Is it true that lim⁡k→∞qk1/k=∞? \lim_{k \to \infty} q_k^{1/k} = \infty? openFormalized
#696No statement retained — open to read what the source holdssolved (Lean)No formal declaration
#697For each mm and α\alpha, the density of the set of integers which are divisible by some d≡1(modm)d \equiv 1 \pmod{m} with 1<d<exp⁡(mα)1 < d < \exp (m ^ \alpha) exists.provedFormalized
#698Is there some h(n)→∞h(n)\to \infty such that for all 2≤i<j≤n/22\leq i<j\leq n/2 gcd((ni),(nj))≥h(n)?\textrm{gcd}\left( \binom{n}{i},\binom{n}{j}\right) \geq h(n)?proved (Lean)Formalized
#699Erdős Problem 699. Is it true that for every 1≤i<j≤n/21 \le i < j \le n / 2 there exists a prime p≥ip \ge i with p∣gcd⁡((ni),(nj))p \mid \gcd\big(\binom{n}{i}, \binom{n}{j}\big)?falsifiableFormalized
#700f n unfolds to the infimum of fSet n.openFormalized
#708No statement retained — open to read what the source holdsopenNo formal declaration
#709No statement retained — open to read what the source holdsopenNo formal declaration
#710No statement retained — open to read what the source holdsopenNo formal declaration
#711No statement retained — open to read what the source holdsopenNo formal declaration
#721No statement retained — open to read what the source holdssolvedNo formal declaration
#726As n→∞n\to \infty ranges over integers ∑p≤n1n∈(p/2,p)(modp)1p∼log⁡log⁡n2\sum_{p\leq n}1_{n\in (p/2,p)\pmod{p}}\frac{1}{p}\sim \frac{\log\log n}{2}?openFormalized
#727Let k≥2k ≥ 2. Does ((n+k)!)2∣(2n)!((n+k)!)^2∣(2n)! hold for infinitely many nn?openFormalized
#728Let ε\varepsilon be sufficiently small and C,C′>0C, C' > 0. Are there integers a,b,na, b, n such that a,b>εna! b!∣n! (a+b−n)!,a, b > \varepsilon n\quad a!\, b! \mid n!\, (a + b - n)!, and Clog⁡n<a+b−n<C′log⁡n?C \log n < a + b - n < C' \log n ?proved (Lean)Formalized
#729Let C>0C>0 be a constant. Are there infinitely many integers a,b,na,b,n with a+b>n+Clog⁡na+b> n+C\log n such that the denominator of n!a!b!\frac{n!}{a!b!}contains only primes ≪C1\ll_C 1?proved (Lean)Formalized
#730Are there infinitely many pairs of integers n<mn < m such that (2nn)\binom{2n}{n} and (2mm)\binom{2m}{m} have the same set of prime divisors?openFormalized
#731No statement retained — open to read what the source holdsopenNo formal declaration
#748No statement retained — open to read what the source holdsprovedNo formal declaration
#763No statement retained — open to read what the source holdsdisprovedNo formal declaration
#764No statement retained — open to read what the source holdsdisprovedNo formal declaration
#768No statement retained — open to read what the source holdsopenNo formal declaration
#769Let c(n)c(n) be minimal such that if k≥c(n)k\geq c(n) then the nn-dimensional unit cube can be decomposed into kk homothetic nn-dimensional cubes. Give good bounds for c(n)c(n) — in particular, is it true that c(n)≫nnc(n)\gg n^n?openFormalized
#770n + 1 is prime iff h n = n + 1. This is described as 'easy to see' in [Er74b].openFormalized
#771No statement retained — open to read what the source holdsprovedNo formal declaration
#772No statement retained — open to read what the source holdsprovedNo formal declaration

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