Browse Problems

Question 1: Is it true that $\limsup M_n = \infty$? Wagner [Wa80] proved that there is some $c > 0$ with $M_n > (\log n)^c$ infintely often. [Wa80] Wagner, Gerold, On a …

Question 2: Is it true that there exists $c > 0$ such that for infinitely many $n$ we have $M_n > n^c$? Beck [Be91] proved that there exists some $c > 0$ …

Question 3: Is it true that there exists $c > 0$ such that, for all large $n$, $\sum_{k \leq n} M_k > n^{1 + c}$?

Question 1: Is it true that $\limsup M_n = \infty$? Wagner [Wa80] proved that there is some $c > 0$ with $M_n > (\log n)^c$ infintely often. [Wa80] Wagner, Gerold, On a …

Question 2: Is it true that there exists $c > 0$ such that for infinitely many $n$ we have $M_n > n^c$? Beck [Be91] proved that there exists some $c > 0$ …

Question 3: Is it true that there exists $c > 0$ such that, for all large $n$, $\sum_{k \leq n} M_k > n^{1 + c}$?

Every odd $1 < n < 10^7$ is the sum of a squarefree number and a power of 2.

Every odd $1 < n < 2^50$ is the sum of a squarefree number and a power of 2.

Suppose that every odd $n$ is the sum of a squarefree number and a power of 2. Then the set of primes $p$ such that $2 ^ p ≡ 2 \mod p …

Erdős often asked this under the weaker assumption that $n > 1$ is not divisible by 4.

Is every odd $n > 1$ the sum of a squarefree number and two powers of 2?

Let $A \subseteq \mathbb{R}$ be an infinite set. Must there be a set $E \subseteq \mathbb{R}$ of positive measure which does not contain any set of the shape $a * A + …

Steinhaus [St20] has proved Erdős 120 to be false whenever $A$ is a finite set.

**Erdős Problem #123** Let $a, b, c$ be three integers which are pairwise coprime. Is every large integer the sum of distinct integers of the form $a^k b^l c^m$ ($k, l, m …

Erdős and Lewin proved this conjecture when $a = 3$, $b = 5$, and $c = 7$. Reference: [ErLe96] Erdős, P. and Lewin, Mordechai, _$d$-complete sequences of integers_. Math. Comp. (1996), 837-840.

A simpler case: the set of numbers of the form $2^k 3^l$ ($k, l ≥ 0$) is d-complete. This was initially conjectured by Erdős in 1992, who called it a "nice and …

A stronger conjecture for numbers of the form $2^k 3^l 5^j$. For any $ε > 0$, all large integers $n$ can be written as the sum of distinct integers $b_1 < ... …

Let $3\leq d_1 < d_2 < \cdots < d_r$ be integers such that all sufficiently large integers can be written as a sum of the shape $\sum_i c_ia_i$ where $c_i \in \{0, …

For any $\varepsilon > 0$, there exists an infinite sequence $2 \le d_0 < d_1 < \dots$ such that all sufficiently large integer can be written as $\sum_{i \in I} a_i$ where …

Let $k \ne 0$ and $3\leq d_1 < d_2 < \cdots < d_r$ be integers of gcd equal to $1$ such that $$\sum_{1 \le i \le r}\frac 1{d_i - 1} \ge 1.$$ …

All sufficiently large integers can be written as $a + b + c$ where $a$ has only the digits $0, 1$ in base $3$, $b$ only the digits $0, 1$ in base …

Let $3 \le d_1 < d_2 < \dots < d_r$ be integers such that $$\sum_{1 \le i \le r}\frac 1{d_i - 1} \ge 1.$$ Can all sufficiently large integers be written as …

Let G be a graph with n vertices such that every subgraph on ≥ $n/2$ vertices has more than $n^2/50$ edges. Must G contain a triangle?

Let $A$ be an infinite set such that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$. Is there such an $A$ with $\liminf \frac{|A …

Let $A$ be an infinite set such that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$. Does there exist some absolute constant $c > …

Let $A$ be an infinite set such that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$. Is it true that $∑_{n \in A} \frac{1}{n} …

Let $A$ be a set of natural numbers with the property that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$. If all elements in …

Given any function $f(x)\to \infty$ as $x\to \infty$ there exists a set $A$ with the property that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > …

Erdős and Sárközy proved that such an $A$ must have density 0. [ErSa70] Erd\H os, P. and Sárk\"ozi, A., On the divisibility properties of sequences of integers. Proc. London Math. Soc. (3) …

An example of an $A$ with the property that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$ and such that \[\liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}\log N …

Let $A$ be a set of natural numbers with the property that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$. If all elements in …

If $A \subseteq \{1, ..., N\}$ is a set with no $a, b, c \in A$ such that $a | (b+c)$ and $a < \min(b,c)$, then $|A| \le N/3 + O(1)$. This …

Let $k\geq 3$. Can the product of any $k$ consecutive integers $N$ ever be powerful? That is, must there always exist a prime $p\mid N$ such that $p^2\nmid N$?

Erdős [Er82c] conjectures that, if $k$ is fixed, then for all $n$ sufficiently large and all positive integers $m$, there must be at least $k$ distinct primes $p$ such that $p\mid m(m+1)\cdots …

Let $k\geq 2$. Erdős and Selfridge [ES75] proved that the product of any $k$ consecutive integers $N$ cannot be a perfect power. [ES75] P. Erdös, J. L. Selfridge, "The product of consecutive …

In [Er80] Erdős asks whether $$ \lim_{k \to \infty} (W(k))^{1/k} = \infty $$

In [Er81] Erdős asks whether $W(k+1) - W(k) \to \infty$.

In [Er80] Erdős asks whether $W(k)/2^k\to \infty$.

When $p$ is prime Berlekamp [Be68] has proved $W(p+1) ≥ p^{2^p}$.

In [Er81] Erdős asks whether $\frac{W(k+1)}{W(k)} \to \infty$.

Gowers [Go01] has proved $$W(k) \leq 2^{2^{2^{2^{2^{k+9}}}}.$$

**Erdős Problem 139**: Let $r_k(N)$ be the size of the largest subset of ${1,...,N}$ which does not contain a non-trivial $k$-term arithmetic progression. Prove that $r_k(N) = o(N)$.

A general version asks, for a fixed $r \in \mathbb{N}$, if a set $A \subseteq \{1, ..., N\}$ has no $a \in A$ and $b_1, ..., b_r \in A$ such that $a …

Let $k≥3$. Are there $k$ consecutive primes in arithmetic progression?

Are there $11$ consecutive primes in arithmetic progression?

The existence of such progressions has been verified for $k≤10$.

Fix a $k \geq 3$. Is it true that there are infinitely many arithmetic prime progressions of length $k$?

It is open, even for $k=3$, whether there are infinitely many such progressions.

Let $s_1 < s_2 < \cdots$ be the sequence of squarefree numbers. Is it true that, for any $\alpha\geq 0$, $$ \lim_{x\to\infty} \frac{1}{x}\sum_{s_n\leq x}(s_{n+1}-s_n)^\alpha $$ exists?