Browse Problems

The set of indices $n$ for which a prime gap is preceeded by a larger or equal prime gap has a natural density of $\frac 1 2$.

There are infintely many indices $n$ such that the prime gap at $n$ is equal to the prime gap at $n+1$. This is equivalent to the existence of infinitely many arithmetic progressions …

The set of indices $n$ for which a prime gap is followed by a larger or equal prime gap has a natural density of $\frac 1 2$.

Are there arbitrarily long arithmetic progressions of primes? Solution: yes. Ref: Green, Ben and Tao, Terence, _The primes contain arbitrarily long arithmetic progressions_

Does there exist, for all large $n$, a polynomial $P$ of degree $n$, with coefficients $\pm1$, such that $$\sqrt n \ll |P(z)| \ll \sqrt n$$ for all $|z|=1$, with the implied constants …

Let $(S_n)_{n \ge 1}$ be a sequence of sets of complex numbers, none of which have a finite limit point. Does there exist an entire transcendental function $f(z)$ such that, for all …

Can every triangle-free graph on $5n$ vertices be made bipartite by deleting at most $n^2$ edges?

A conjecture by Heath-Brown: The sum of squares of the first $N$ gaps between consecutive primes behaves like $N * (log N)^2$.

Cramér proved an upper bound of $O(N(\log N)^4)$ conditional on the Riemann hypothesis.

Is 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 …

Let $f(n)$ count the number of solutions to $n=p+2^k$ for prime $p$ and $k\geq 0$. Show that $f(n)=o(\log n)$.

Let `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 …

It is well-known that the conjecture above is true when `c₁` is sufficiently small.

Let $f:\mathbb{N}\to \{-1,1\}$ be a multiplicative function. Is it true that \[ \lim_{N\to \infty}\frac{1}{N}\sum_{n\leq N}f(n)\] always exists? The answer is yes, as proved by Wirsing [Wi67], and generalised by Halász [Ha68].

For every $n>2$ there exist distinct integers $1 ≤ x < y < z$ such that $\frac 4 n = \frac 1 x + \frac 1 y + \frac 1 z$.

For any fixed $a$, if $n$ is sufficiently large in terms of $a$ then there exist distinct integers $1 ≤ x < y < z$ such that $\frac a n = \frac …

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 = …

Let $C > 1$. Does the set of integers of the form $p + \lfloor C^k \rfloor$, for some prime $p$ and $k\geq 0$, have density $>0$?

Romanoff [Ro34] proved that the answer is yes if $C$ is an integer. [Ro34] Romanoff, N. P., _Über einige Sätze der additiven Zahlentheorie_. Math. Ann. (1934), 668-678.

Let $A\subseteq\mathbb{N}$ be an infinite set such that $|A\cap \{1, ..., N\}| = o(N)$. Is it true that $$ \limsup_{N\to\infty}\frac{|(A + A)\cap \{1, ..., N\}|}{|A \cap \{1, ..., N\}|} \geq 3? $$ …

Let $A\subseteq\mathbb{N}$ be an infinite set such that $|A\cap \{1, ..., N\}| = o(N)$. Then $$ \limsup_{N\to\infty}\frac{|(A + A)\cap \{1, ..., N\}|}{|A \cap \{1, ..., N\}|} \geq 2. $$

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?

Erdős proved the answer is yes under the stronger condition that $\limsup \frac{n_k}{k^t} = \infty$ for all $t\geq 1$. [ErGr80] Erdős, P. and Graham, R., _Old and new problems and results in …

Are there infinitely many $n$ such that $\omega(n + k) \ll k$ for all $k \geq 1$? Here $\omega(n)$ is the number of distinct prime divisors of $n$.

Is $$\sum_{n} \frac{\phi(n)}{2^n}$$ irrational? Here $\phi$ is the Euler totient function.

Let $n_1 < n_2 < \dots$ be an arbitrary sequence of integers, each with an associated residue class $a_i \pmod{n_i}$. Let $A$ be the set of integers $n$ such that for every …

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 ($p_1=2, p_2=3, \dots$).

`∑ σ 0 n / n!` is irrational. This is proved in [ErSt71].

`∑ σ 1 n / n!` is irrational. This is proved in [ErSt74].

`∑ σ 2 n / n!` is irrational. This is proved in [ErKa54].

`∑ σ 3 n / n!` is irrational. This is proved in [ScPu06] and [FLC07].

`∑ σ 4 n / n!` is irrational. This is proved in [Pr22].

For a fixed `k ≥ 5`, is `∑ σ k n / n!` irrational?.

If the prime `k`-tuples conjecture is true, then `∑ σ k n / n!` is irrational. This is proved in [FLC07].

If Schinzel's conjecture is true, then `∑ σ k n / n!` is irrational for all `k`. This is proved in [ScPu06].

`∑ σ 4 n / n!` is irrational. This is proved in [Pr22].

`∑ σ 1 n / n!` is irrational. This is proved in [ErSt74].

`∑ σ 3 n / n!` is irrational. This is proved in [ScPu06] and [FLC07].

`∑ σ 2 n / n!` is irrational. This is proved in [ErKa54].

`∑ σ 0 n / n!` is irrational. This is proved in [ErSt71].

For a fixed `k ≥ 5`, is `∑ σ k n / n!` irrational?.

If the prime `k`-tuples conjecture is true, then `∑ σ k n / n!` is irrational. This is proved in [FLC07].

If Schinzel's conjecture is true, then `∑ σ k n / n!` is irrational for all `k`. This is proved in [ScPu06].

Let $a_1 < a_2 < \dotsc$ be an infinite sequence of positive integers such that $\frac{a_{i+1}}{a_i} \to 1$. If every arithmetic progression contains infinitely many integers which are the sum of distinct …

Let $A\subseteq\mathbb{N}$ be an infinite set. Is $$ \sum_{n\in A} \frac{1}{2^n - 1} $$ irrational?

Show that $$ \sum_{n} \frac{d(n)}{2^n} $$ is irrational. [Er48] Erdős, P., _On arithmetical properties of Lambert series_. J. Indian Math. Soc. (N.S.) (1948), 63-66.

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 $n$?

Is $\sum_n \frac{d(n)}{t^n}$ irrational, where $t ≥ 2$ is an integer. Solution: True (proved by Erdős, see Erdős Problems website)