Browse Problems

Let $\epsilon > 0$ be given. Then, for a sequence of sufficiently large consecutive primes $p_1 < \cdots < p_k$ such that $$ \sum_{i=1}^k \frac{1}{p_i} < 1 < \sum_{i=1}^{k + 1} \frac{1}{p_i}, …

A problem of Erdős and Sárközy who proved $$ g_3(n) \gg \frac{3^n}{n^{O(1)}}. $$

$F(n) / \log n \to \infty as n \to \infty$

Does the set of integers of the form $n + \varphi(n)$ have positive (lower) density? [GIL24] proved this was true.

Is there an absolute constant $C > 0$ such that every integer $n$ with $\sigma(n) > Cn$ is the distinct sum of proper divisors of $n$? This has been solved in the …

Show that if the constant $C > 0$ is such that every integer $n$ with $\sigma(n) > Cn$ is the distinct sum of proper divisors of $n$, then we must have $C …

Are there infinitely many $n$ such that, for all $k\geq 1$ $$ \tau(n + k) \ll k? $$

Is it true that, for any $a \in \mathbb{Z}$, there are infinitely many $n$ such that $$\phi(n) | n + a$$?

When $n > 1$, Lehmer conjectured that $\phi(n) | n - 1$ if and only if $n$ is prime.

$F(n) \le O(n^{1/2} \ln ^ {3/4} n)$ Theorem 1.4 from [AKS07] [AKS07] Alon, N. and Krivelevich, M. and Sudakov, B., Large nearly regular induced subgraphs. arXiv:0710.2106 (2007).

**Erdos Problem 830, Part 1** We say that $a,b\in \mathbb{N}$ are an amicable pair if $\sigma(a)=\sigma(b)=a+b$. Are there infinitely many amicable pairs?

**Erdos Problem 830, Part 2** We say that $a,b\in \mathbb{N}$ are an amicable pair if $\sigma(a)=\sigma(b)=a+b$. If $A(x)$ counts the number of amicable $1\leq a\leq b\leq x$ then is it true that …

We say that $a,b\in \mathbb{N}$ are an amicable pair if $\sigma(a)=\sigma(b)=a+b$. If $A(x)$ counts the number of amicable $1\leq a\leq b\leq x$ then one can show that $A(x) = o(x)$.

We say that $a,b\in \mathbb{N}$ are an amicable pair if $\sigma(a)=\sigma(b)=a+b$. If $A(x)$ counts the number of amicable $1\leq a\leq b\leq x$ then one can show that $A(x) \leq x \exp(-(\log x)^{1/3})$.

We say that $a,b\in \mathbb{N}$ are an amicable pair if $\sigma(a)=\sigma(b)=a+b$. If $A(x)$ counts the number of amicable $1\leq a\leq b\leq x$ then one can show that $A(x) \leq x \exp(-(\tfrac{1}{2}+o(1))(\log x\log\log …

Does there exist a $k>2$ such that the $k$-sized subsets of {1,...,2k} can be coloured with $k+1$ colours such that for every $A\subset \{1,\ldots,2k\}$ with $\lvert A\rvert=k+1$ all $k+1$ colours appear among …

Alternative statement of Erdős Problem 835 using the chromatic number of the Johnson graph. This is equivalent to asking whether there exists $k > 2$ such that the chromatic number of the …

Let $C > 0$. Is it true that the set of integers of the form $n = b_1 + \cdots + b_t$, with $b_1 < \cdots < b_t$, where $b_i = 2^{k_i}3^{l_i}$ …

**Erdős Problem 846** Let `A ⊂ ℝ²` be an infinite set for which there exists some `ϵ>0` such that in any subset of `A` of size `n` there are always at least …

Let $A \subset \mathbb{N}$ be an infinite set for which there exists some $\epsilon > 0$ such that in any subset of $A$ of size $n$ there is a subset of size …

Is the maximum size of a set $A ⊆ \{1, \dots, N\}$ such that $ab + 1$ is never squarefree (for all $a, b ∈ A$) achieved by taking those $n ≡ …

There exists $N₀$ such that for all $N ≥ N₀$, if $A ⊆ \{1, \dots, N\}$ satisfies that $ab + 1$ is never squarefree for all $a, b ∈ A$, then $|A| …

Is it true that, for every integer $t\geq1$, there is some integer $a$ such that ${n \choose k} = a$ with $1\leq k \le \frac{n}{2}$ has exactly $t$ solutions?

Is it true that, for all large $n$, $f(n + 1) \ge f(n)$?

Can there exist two distinct integers $x$ and $y$ such that $x,y$ have the same prime factors, $x+1,y+1$ have the same prime factors, and $x+2,y+2$ also have the same prime factors?

Let $\epsilon > 0$. Is there some $r \ll_\epsilon 1$ such that the density of integers of the form $2^k+n$, where $k \geq 0$ and $n$ has at most $r$ prime divisors, …

The set of integers of the form `2^k+p` (where `p` is prime) has positive lower density. Formalisation note: here we also allow `p = 1` since this simplifies the code and is …

Let $d_n = p_{n+1} - p_n$, where $p_n$ is the $n$th prime. Let $r(x)$ be the smallest even integer $t$ such that $d_n = t$ has no solutions for $n \le x$. …

Let $d_n = p_{n+1} - p_n$, where $p_n$ is the $n$th prime. Let $r(x)$ be the smallest even integer $t$ such that $d_n = t$ has no solutions for $n \le x$. …

A weaker version of the problem proved by Erdos: The density `dₜ` of `DivisorSumSet (t : ℕ)` is bounded from below by `1 / log (t) ^ c₃` and from above by …

A weaker version of the problem proved by Erdos: The density `dₜ` of `DivisorSumSet (t : ℕ)` is bounded from below by `1 / log (t) ^ c₃` and from above by …

There exists a constant $C>0$ such that, for all large $N$, if $A\subseteq \{1,\ldots,N\}$ has size at least $\frac{5}{8}N+C$ then there are distinct $a,b,c\in A$ such that $a+b,a+c,b+c\in A$. A problem of …

It is a classical folklore fact that if $A\subseteq \{1,\ldots,2N\}$ has size $\geq N+2$ then there are distinct $a,b\in A$ such that $a+b\in A$, which establishes the $k=2$ case.

Erdős and Sós conjectured that $f_k(N)\sim \frac{1}{2}\left(1+\sum_{1\leq r\leq k-2}\frac{1}{4^r}\right) N$, where $f_k(N)$ is the minimal size of a subset of $\{1, \dots, N\}$ guaranteeing $k$ elements have all pairwise sums in the …

Choi, Erdős, and Szemerédi [CES75] have proved that, for all $k\geq 3$, there exists $\epsilon_k>0$ such that (for large enough $N$) $f_k(N)\leq \left(\frac{2}{3}-\epsilon_k\right)N$.

Let $A$ be an additive basis of order $2$, let $f(n)$ denote the number of ways in which $n$ can be written as the sum of two elements from $A$. If $f(n) …

Let $A$ be an additive basis of order $2$, let $f(n)$ denote the number of ways in which $n$ can be written as the sum of two elements from $A$. If $f(n) …

Erdős and Nathanson proved that this is true if $f(n) > (\log \frac{4}{3})^{-1} \log n$ for all large $n$.

Härtter and Nathanson proved that there exist additive bases which do not contain any minimal additive bases.

Let $A = \{a_1 < a_2 < \dots\} \subseteq \mathbb{N}$ and let $F(A,X,k)$ count the number of $i$ such that $[a_i,a_{i+1}, \dots ,a_{i+k−1}] < X$, where the left-hand side is the least …

Let `A ⊂ ℕ` be an additive basis of order `k` which is minimal in the sense that if `B ⊂ A` is any infinite set, then `A \ B` is not …

Is it true that, for every $k \geq 1$, there exist integers $N_1 < \dots < N_k$ such that $|\cap_i D(N_i)| \geq k$?

Erdős and Rosenfeld [ErRo97] proved this is true for $k=2$.

Jiménez-Urroz [Ji99] proved this for $k=3$.

Bremner [Br19] proved this for $k=4$.

Let $\epsilon>0$. Is it true that, for all large $n$, the number of divisors of $n$ in $(n^{1/2},n^{1/2}+n^{1/2-\epsilon})$ is $O_\epsilon(1)$? Erdős attributes this conjecture to Ruzsa.

Erdős and Rosenfeld [ErRo97] proved that, for any constant $C>0$, all large $n$ have at most $1+C^2$ many divisors in $[n^{1/2}, n^{1/2}+Cn^{1/4}]$.

Erdős and Rosenfeld [ErRo97] proved that there are infinitely many $n$ such that there are four divisors of $n$ in $(n^{1/2},n^{1/2}+16n^{1/4})$.