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.
Is \[\sum_{n=2}^\infty \frac{1}{n!-1}\] irrational?
Let \(a_1 < a_2 < \dots\) be a sequence of integers such that \(\lim_{n\to\infty} \frac{a_n}{a_{n-1}^2} = 1\) and \(\sum \frac{1}{a_n} \in \mathbb{Q}\). Then, for all sufficiently large \(n \ge 1\), \(a_n = a_{n-1}^2 - a_{n-1} + 1\).
Let \(n_1 < n_2 < \cdots\) be a sequence of integers such that \[ \limsup \frac{n_k}{k} = \infty. \] Is \[ \sum_{k=1}^{\infty} \frac{1}{2^{n_k}} \] transcendental?
Is \[\sum_{n} \frac{\phi(n)}{2^n}\] irrational? Here \(\phi\) is the Euler totient function.
Is \[ \sum_{n=1}^\infty \frac{\sigma(n)}{2^n} \] irrational? Here \(\sigma(n)\) is the sum of divisors function. The answer is yes, as shown by Nesterenko [Ne96]. [Ne96] Nesterenko, Yu V., _Modular functions and transcendence questions_,…
Is \(\sum_{n=1}^\infty \frac{p_n}{2^n}\) irrational? Here \(p_n\) is the \(n\)-th prime (\(p_1=2, p_2=3, \dots\)).
Erdős Problem 252: irrationality of the sum for a given \(k\).
Let \(A\subseteq\mathbb{N}\) be an infinite set. Is \[ \sum_{n\in A} \frac{1}{2^n - 1} \] irrational?
Let \(a_n \to \infty\) be a sequence of non-zero natural numbers. Is \(\sum_n \frac{d(n)}{(a_1 ... a_n)}\) irrational, where \(d(n)\) is the number of divisors of \(n\)? This was proved affirmatively by Chojecki and GPT-5.4 Pro [Ch26], and…
Is \(\sum_{n} \mu(n)^2\frac{n}{2^n}\) irrational? This is true, and was proved by Chen and Ruzsa. [ChRu99] Chen, Yong-Gao and Ruzsa, Imre Z., On the irrationality of certain series. Period. Math. Hungar. (1999), 31--37.
Let \(a_1 < a_2 < \cdots\) be an increasing sequence such that \(\frac{a_n}{n} \to \infty\). Is the sum \(\sum_{n}^{\infty} \frac{a_n}{2^{a_n}}\) irrational? For a proof, see [Wang, *Sparse Polynomial-Weighted Expansions*]…
Is \(a_n = 2^{2^n}\) an irrationality sequence in the above sense?
Is \(2^n\) an example of an irrationality sequence? Kovač and Tao proved that it is not [KoTa24] [KoTa24] Kovač, V. and Tao T., On several irrationality problems for Ahmes series. arXiv:2406.17593 (2024).
ambiguous statement
Let \(a_n\) be an infinite sequence of positive integers such that \(\sum \frac{1}{a_n}\) converges. There exists some integer \(t \ge 1\) such that \(\sum \frac{1}{a_n + t}\) is irrational. This was disproven by Kovač and Tao in [KoTa24].…
Let \(F_1=F_2=1\) and \(F_{n+1} = F_n + F_{n-1}\) be the Fibonacci sequence. Let \(n_1 < n_2 < \dots\) be an infinite sequence with \(\frac{n_{k+1}}{n_k} \ge c > 1\). Must \(\sum_k \frac 1 {F_{n_k}}\) be irrational?
Let \(f(n)\to \infty\) as \(n\to \infty\). Is it true that \[\sum_{n\geq 1} \frac{1}{(n+1)\cdots (n+f(n))}\] is irrational? Erdős and Graham [ErGr80] write 'the answer is almost surely in the affirmative if \(f(n)\) is assumed to be…
Let \(t>1\) be a rational number. Is \(\sum_{n=1}^\infty\frac{1}{t^n-1}=\sum_{n=1}^\infty \frac{\tau(n)}{t^n}\) irrational, where \(\tau(n)\) counts the divisors of \(n\)? A conjecture of Chowla.
**Erdős Problem 1050.** The series \(\sum_{n=1}^\infty \frac{1}{2^n - 3}\) is irrational. The source sum runs over \(n \ge 1\); as a tsum over ℕ (which starts at \(0\)) it is reindexed \(n \mapsto n + 1\), i.e.…
Is it true that if \(a_0 < a_1 < a_2 < \cdots\) is a strictly increasing sequence of integers with \(\liminf a_n^{1/2^n} > 1\), then the series \(\sum_{n=0}^\infty \frac{1}{a_n \cdot a_{n+1}}\) is irrational? This was solved in the…