Browse Problems

Let $a_n \to \infty$ be a monotone 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 $n$? Solution: True (proved by Erdős …

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 $A\subset\mathbb{N}$ be infinite such that $\sum_{a \in A} \frac{1}{a} = \infty$. Must there exist some $k\geq 1$ such that almost all integers have a divisor of the form $a+k$ for some …

Is $a_n = 2^{2^n}$ an irrationality sequence in the above sense?

Must every irrationality sequence $a_n$ in the above sense satisfy $a_n^{1/n} \to \infty$ as $n \to \infty$?

Koizumi [Ko25] showed that $a_n = \lfloor \alpha^{2^n} \rfloor$ is an irrationality sequence for all but countably many $\alpha > 1$. [Ko25] Koizumi, J., Irrationality of the reciprocal sum of doubly exponential …

A folklore result states that any $a_n$ satisfying $\lim_{n \to \infty} a_n^{\frac{1}{2^n}} = \infty$ has $\sum \frac{1}{a_n}$ converging to an irrational number.

Kovač and Tao [KoTa24] proved that any strictly increasing sequence $a_n$ such that $\sum \frac{1}{a_n}$ converges and $\lim \frac{a_{n+1}}{a_n^2} = 0$ is not an irrationality sequence in the above sense. [KoTa24] Kovač, …

On the other hand, if there exists some $\varepsilon > 0$ such that $a_n$ satisfies $\liminf \frac{a_{n+1}}{a_n^{2+\varepsilon}} > 0$, then $a_n$ is an irrationality sequence by the above folklore result `erdos_263.variants.folklore`.

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).

Is $n!$ an example of an irrationality sequence?

One example is $2^{2^n}$.

Kovač and Tao [KoTa24] generally proved that any strictly increasing sequence of positive integers $a_n$ such that $\sum \frac{1}{a_n}$ converges and $$ \liminf_{n \to \infty} (a_n^2 \sum_{k > n} \frac{1}{a_k^2}) > 0 …

On the other hand, Kovač and Tao [KoTa24] do prove that for any function $F$ with $\lim_{n \to \infty} \frac{F(n + 1)}{F(n)} = \infty$ there exists such an irrationality sequence with $a_n …

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 …

In fact, Kovač and Tao proved in [KoTa24] that there exists a strictly increasing sequence $a_n$ of positive integers such that $\sum \frac{1}{a_n + t}$ converges to a rational number for all …

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 …

The sum $\sum_n \frac 1 {F_{n}}$ itself was proved to be irrational by André-Jeannin. Ref: André-Jeannin, Richard, _Irrationalité de la somme des inverses de certaines suites récurrentes_.

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 $\frac {n_k}{k} \to \infty$. Must $\sum_k \frac 1 {F_{n_k}}$ …

Good [Go74] and Bicknell and Hoggatt [BiHo76] have shown that $\sum_n \frac 1 {F_{2^n}}$ is irrational. Ref: * [Go74] Good, I. J., _A reciprocal series of Fibonacci numbers_ * [BiHo76] Hoggatt, Jr., …

Let `X` be the set of points in `Fin d → ℝ` of the shape `fun i : Fin d => ∑' n : A, (1 : ℝ) / (n + i)` …

This theorem addresses the case where the set of primes $P$ is infinite. In this case the sum is irrational.

Let $P$ be a finite set of primes with $|P| \ge 2$ and let $\{a_1 < a_2 < \dots\}$ be the set of positive integers whose prime factors are all in $P$. …

Let $P$ be a finite set of primes with $|P| \ge 2$ and let $\{a_1 < a_2 < \dots\}$ be the set of positive integers whose prime factors are all in $P$. …

If we allow for $\sum_{a\in A} \frac{1}{a} < \infty$ then Rusza has found a counter-example.

Tenenbaum asked the weaker variant where for every $\epsilon>0$ there is some $k=k(\epsilon)$ such that at least $1-\epsilon$ density of all integers have a divisor of the form $a+k$ for some $a\in …

Is there a covering system all of whose moduli are of the form $p-1$ for some primes $p \geq 5$?

Is there a covering system all of whose moduli are of the form $p-1$ for some primes $p \geq 3$?

If `G` is a group then can there exist an exact covering of `G` by more than one cosets of different sizes? (i.e. each element is contained in exactly one of the …

If `G` is a finite abelian group then there cannot exist an exact covering of `G` by more than one cosets of different sizes? (i.e. each element is contained in exactly one …

If a finite system of $r$ congruences $\{ a_i\pmod{n_i} : 1\leq i\leq r\}$ (the $n_i$ are not necessarily distinct) covers $2^r$ consecutive integers then it covers all integers. This is best possible …

Is there an infinite Lucas sequence $a_0, a_1, \ldots$ where $a_{n+2} = a_{n+1} + a_n$ for $n \ge 0$ such that all $a_k$ are composite, and yet no integer has a common …

Is it true that, for every $c$, there exists an $n$ such that $\sigma(n)>cn$ but there is no covering system whose moduli all divide $n$? This was answered affirmatively by Haight [Ha79].

Let $p\colon \mathbb{Z} \rightarrow \mathbb{Z}$ be a polynomial whose leading coefficient is positive and such that there exists no $d≥2$ with $d ∣ p(n)$ for all $n≥1$. Is it true that, for …

Graham [Gr63] has proved this when $p(x)=x$.

Let $f(k)$ be the minimal value of $n_k$ such that there exist $n_1 < n_2 < \dots < n_k$ with $$ 1 = \frac{1}{n_1} + \cdots + \frac{1}{n_k}. $$ Is it true …

It is trivial that $f(k)\geq (1 + o(1)) \frac{e}{e - 1}k$.

Is it true that there are only finitely many pairs of intervals $I_1$, $I_2$ such that $$ \sum_{n_1 \in I_1} \frac{1}{n_1} + \sum_{n_2 \in I_2} \frac{1}{n_2} \in \mathbb{N}? $$

Is it true for any $k > 2$ that only finitely many $k$ intervals satisfy this condition?

This is still open even if $|I_2| = 1$.

It is perhaps true with two intervals replaced by any $k$ intervals.

Is it true that, for all sufficiently large $k$, there exists finite intervals $I_1, \dotsc, I_k \subset \mathbb{N}$ with $|I_i| \geq 2$ for $1 \leq i \leq k$ such that $$ 1 …

Let $k(N)$ denote the smallest $k$ such that there exists $N ≤ n_1 < ⋯ < n_k$ with $\frac 1 {n_1} + ... + \frac 1 {n_k} = 1$ Is it true …

Erdős and Straus have proved the existence of some constant $c>0$ such that $-c < k(N)-(e-1)N \ll \frac N {\log N}$

Does every set $A \subseteq \mathbb{N}$ of positive density contain some finite $S \subset A$ such that $\sum_{n \in S} \frac{1}{n} = 1$? The answer is yes, proved by Bloom [Bl21]. [Bl21] …

In [Bl21] it is proved under the weaker assumption that `A` only has positive upper density.

Is there an infinite sequence $a_1 < a_2 < \dots$ such that $a_{i+1} - a_i = O(1)$ and no finite sum of $\frac{1}{a_i}$ is equal to 1?

The corresponding question is also false if one replaces sequences such that $a_{i+1} - a_i = O(1)$ with sets of positive density, as follows from [Bl21]. The statement is as follows: If …

Is it true that, for every $\varepsilon > 0$, $h(N) = \sqrt N + O_{\varespilon}(N^\varespilon)

Is it true that in any finite colouring of the integers there exists a monochromatic solution to $\frac 1 a = \frac 1 b + \frac 1 c$ with distinct $a, b, …