Browse Problems

The density of hyperbolicity conjecture, stating that the set of all parameters `c` for which `fun z ↦ z ^ 2 + c` has an attracting cycle is dense in the Mandelbrot …

The density of hyperbolicity conjecture for Multibrot sets, stating that the set of all parameters `c` for which `fun z ↦ z ^ n + c` has an attracting cycle is dense …

Does the determinant of the sum $A + B$ of two $n \times n$ normal complex matrices $A$ and $B$ always lie in the convex hull of the $n!$ points $\prod_i (\lambda(A)_i …

**Dickson's conjecture** If a finite set of in linear integer forms $f_i(n) = a_i n+b_i$ satisfies Schinzel condition, there exist infinitely many natural numbers $m$ such that $f_i(m)$ are primes for all …

The dihedral group `DihedralGroup n` (of order `2n`) is a Leinster group if and only if `n` is an odd perfect number. This gives a one-to-one correspondence between dihedral Leinster groups and …

Any T2, Toronto space is discrete.

Johnson's bound for the chromatic number of the Johnson graph.

Does there exist a constant $c > 0$ so that for every graph $G$ and every $\epsilon$ between $0$ and $1$, $V$ contains an $\epsilon$-light subset $S$ of size at least $c …

Is there some $k$ such that every integer is the sum of a prime and at most $k$ powers of $2$?

Is the diameter of $A$ at least $Cn$ for some constant $C > 0$?

Are there infinitely many solutions to $\phi(n) = \phi(n+1)$, where $\phi$ is the Euler totient function?

Erdős, Pomerance, and Sárközy [EPS87] proved that for all large $x$, the number of $n \leq x$ with $\phi(n) = \phi(n+1)$ is at most $$\frac{x}{\exp((\log x)^{1/3})}$$. [EPS87] Erd\H os, Paul and Pomerance, …

Erdős [Er85e] says that, presumably, for every $k \geq 1$ the equation $$\phi(n) = \phi(n+1) = \cdots = \phi (n+k)$$ has infinitely many solutions. [Er85e] Erdős, P., _Some problems and results in …

For any fixed c > 0, if x is sufficiently large then there exists n ≤ x such that the values of φ(n+k) are all distinct for 1 ≤ k ≤ (log …

Erdős, Pomerance, and Sárközy [EPS87] proved that if φ(n+k) are all distinct for 1 ≤ k ≤ K then K ≤ n / exp(c (log n)^{1/3}) for some constant c > 0. …

From [GuKa15]: diameter $\gg n / \log n$.

From [Kanold]: diameter $\geq n^{3/4}$. TODO: find reference

From [Piepmeyer]: 9 points with diameter $< 5$. TODO: find reference

Stronger conjecture: diameter $\geq n - 1$ for sufficiently large $n$.

What is the infimum of `|{x ∈ ℝ : |f x| < 1}|` over all nonconstant monic polynomials `f` such that all of its roots are real and contained in `[-1,1]`?

The infimum of `|{x ∈ ℝ : |f x| < 1}|` over all nonconstant monic polynomials `f` such that all of its roots are real and contained in `[-1,1]` is `≥ 2 …

The infimum of `|{x ∈ ℝ : |f x| < 1}|` over all nonconstant monic polynomials `f` such that all of its roots are real and contained in `[-1,1]` is `< 1.835`.

The supremum of `|{x ∈ ℝ : |f x| < 1}|` over all monic polynomials `f` such that all of its roots are real and contained in `[-1,1]` is `2 * 2 …

Let $$ f(z) = \prod_{i=1}^{n} (z - z_i) \in \mathbb{C}[x] $$ with $|z_i| < 1$ for all $i$. Conjecture: Must there always exist a path of length less than 2 in $$ …

**Erdős Problem 1043**: Let $f\in \mathbb{C}[x]$ be a monic polynomial. Must there exist a straight line $\ell$ such that the projection of \[\{ z: \lvert f(z)\rvert\leq 1\}\] onto $\ell$ has measure at …

On the other hand, Pommerenke also proved there always exists a line such that the projection has measure at most 3.3.

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 [Er48] proved that this is true if $t\geq 2$ is an integer.

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 …

Erdős [Er88c] notes that if the sequence grows rapidly to infinity (specifically, if $a_{n+1} \geq C \cdot a_n^2$ for some constant $C > 0$), then the series is irrational.

Are there only finitely many unitary perfect numbers?

Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Is it true that $f(n)=o(n)$?

Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Is it true that $f(n)=o(n)$ for almost all …

Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Is it true that $\limsup f(n)/n=\infty$?

A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of …

A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of …

A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of …

Let $k ≥ 2$. Does there exist a prime $p$ and consecutive intervals $I_0,\dots,I_k$ such that $\prod\limits_{n{\in}I_i}n \equiv 1 \mod n$ for all $1 \le i \le k$?

Are there infinitely many primes $p$ such that $p - k!$ is composite for each $k$ such that $1 ≤ k! < p$?

The conjecture is about the function $f(n)$ which counts the number of solutions to $k\sigma(k)=n$, where $\sigma(k)$ is the sum of divisors of $k$. The first bound is that $f(n)$ grows slower …

How many (ordered) solutions are there to `σ(a) + σ(b) = σ(a + b)` with `a + b ≤ x`? Is it true that this number is asymptotic to `c * x` …

Lebensold proved that for large `n`, the function `f n` lies between `0.6725 n` and `0.6736 n`.

Erdős asked whether the limiting density `f n / n` exists and, if so, whether it is irrational.

The interval `[⌊n/3⌋, n]` is fork-free, and therefore `f n` is at least `⌈2n / 3⌉`.

Estimate $n_k$ by finding a better upper bound.

[Cambie observed](https://www.erdosproblems.com/1063) the improved bound $n_k \le k \cdot \operatorname{lcm}(1, \dotsc, k - 1)$.

Erdős and Selfridge noted that, for $n \ge 2k$ with $k \ge 2$, at least one of the numbers $n - i$ for $0 \le i < k$ fails to divide $\binom{n}{k}$ …