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

22 Problems

SelectNumberQuestionOpen
#68Is ∑n=2∞1n!−1\sum_{n=2}^\infty \frac{1}{n!-1} irrational?Source says openFormalized
#69Is ∑n≥2ω(n)2n \sum_{n\geq 2}\frac{\omega(n)}{2^n} irrational? (Here ω(n)\omega(n) counts the number of distinct prime divisors of nn.)Source says provedFormalized
#247Let n1<n2<⋯n_1 < n_2 < \cdots be a sequence of integers such that lim sup⁡nkk=∞. \limsup \frac{n_k}{k} = \infty. Is ∑k=1∞12nk \sum_{k=1}^{\infty} \frac{1}{2^{n_k}} transcendental?Source says openFormalized
#249Is ∑nϕ(n)2n\sum_{n} \frac{\phi(n)}{2^n} irrational? Here ϕ\phi is the Euler totient function.Source says openFormalized
#250Is ∑n=1∞σ(n)2n \sum_{n=1}^\infty \frac{\sigma(n)}{2^n} irrational? Here σ(n)\sigma(n) is the sum of divisors function.Source says provedFormalized
#251Is ∑n=1∞pn2n\sum_{n=1}^\infty \frac{p_n}{2^n} irrational? Here pnp_n is the nn-th prime (p1=2,p2=3,…p_1=2, p_2=3, \dots).Source says openFormalized
#252Erdős Problem 252: irrationality of the sum for a given kk.Source says openFormalized
#257Let A⊆NA\subseteq\mathbb{N} be an infinite set. Is ∑n∈A12n−1 \sum_{n\in A} \frac{1}{2^n - 1} irrational?Source says openFormalized
#258Let an→∞a_n \to \infty be a sequence of non-zero natural numbers. Is ∑nd(n)(a1...an)\sum_n \frac{d(n)}{(a_1 ... a_n)} irrational, where d(n)d(n) is the number of divisors of nn?Source says proved (Lean)Formalized
#259Is ∑nμ(n)2n2n\sum_{n} \mu(n)^2\frac{n}{2^n} irrational?Source says proved (Lean)Formalized
#260Let a1<a2<⋯a_1 < a_2 < \cdots be an increasing sequence such that ann→∞\frac{a_n}{n} \to \infty. Is the sum ∑n∞an2an\sum_{n}^{\infty} \frac{a_n}{2^{a_n}} irrational?Source says openFormalized
#262No statement retained — open to read what the source holdsSource says solvedNo formal declaration
#263Is an=22na_n = 2^{2^n} an irrationality sequence in the above sense?Source says openFormalized
#264Is 2n2^n an example of an irrationality sequence? Kovač and Tao proved that it is not [KoTa24]Source says openFormalized
#265No statement retained — open to read what the source holdsSource says openNo formal declaration
#266Let ana_n be an infinite sequence of positive integers such that ∑1an\sum \frac{1}{a_n} converges. There exists some integer t≥1t \ge 1 such that ∑1an+t\sum \frac{1}{a_n + t} is irrational.Source says disprovedFormalized
#267Let F1=F2=1F_1=F_2=1 and Fn+1=Fn+Fn−1F_{n+1} = F_n + F_{n-1} be the Fibonacci sequence. Let n1<n2<…n_1 < n_2 < \dots be an infinite sequence with nk+1nk≥c>1\frac{n_{k+1}}{n_k} \ge c > 1. Must ∑k1Fnk\sum_k \frac 1 {F_{n_k}} be irrational?Source says openFormalized
#269This theorem addresses the case where the set of primes PP is infinite. In this case the sum is irrational.Source says openFormalized
#270No statement retained — open to read what the source holdsSource says disprovedNo formal declaration
#1049Let t>1t>1 be a rational number. Is ∑n=1∞1tn−1=∑n=1∞τ(n)tn\sum_{n=1}^\infty\frac{1}{t^n-1}=\sum_{n=1}^\infty \frac{\tau(n)}{t^n} irrational, where τ(n)\tau(n) counts the divisors of nn?Source says openFormalized
#1050No statement retained — open to read what the source holdsSource says provedNo formal declaration
#1051Is it true that if a0<a1<a2<⋯a_0 < a_1 < a_2 < \cdots is a strictly increasing sequence of integers with lim inf⁡an1/2n>1\liminf a_n^{1/2^n} > 1, then the series ∑n=0∞1an⋅an+1\sum_{n=0}^\infty \frac{1}{a_n \cdot a_{n+1}} is irrational?Source says proved (Lean)Formalized