Browse Problems

The following weaker version of the mean value problem has been proven. Given a complex polynomial $p$ of degree $d ≥ 2$ and a complex number $z$ there a critical point $c$ …

The following tighter bound depending on the degree $d$ of the polynomial $p$, in the case of $p$ only having real roots has been shown by Tischler. $|p(z)-p(c)|/|z-c| \le (d-1)/d \cdot |p'(z)|$

The following tighter bound depending on the degree $d$ of the polynomial $p$, in the case of $p$ all roots having the same norm has been shown by Tischler. $|p(z) - p(c)|/|z-c| …

A lower bound of $\sqrt{4 - \sqrt{15}}$.

A lower bound of $1 - frac{1}{\sqrt 2}$. Scherk (written communication), see [On the minimal overlap problem of Erdös](https://eudml.org/doc/206397) by *Leo Moser*, Аста Аrithmetica V, p. 117-119, 1959

A lower bound of $\frac{4 - \sqrt{6}}{5}. See [On the intersection of a linear set with the translation of its complement](https://bibliotekanauki.pl/articles/969027) by *Stanisław Świerczkowski1*, Colloquium Mathematicum 5(2), p. 185-197, 1958

A lower bound of $0.379005$. See [Erdős' minimum overlap problem](https://arxiv.org/abs/2201.05704) by *Ethan Patrick White*, 2022

The example (with $N$ even), $A = \{\frac N 2 + 1, \dots, \frac{3N}{2}\}$ shows an upper bound of $\frac 1 2$.

An upper bound of $0.38200298812318988$. See [Advances in the Minimum Overlap Problem](https://doi.org/10.1006%2Fjnth.1996.0064) by *Jan Kristian Haugland*, Journal of Number Theory Volume 58, Issue 1, p 71-78, 1996

An upper bound of $0.3809268534330870$. See [The minimum overlap problem](https://www.neutreeko.net/mop/index.htm) by *Jan Kristian Haugland*

An upper bound of $\frac 2 5$. See [Minimal overlapping under translation.](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-62/issue-6) by *T. S. Motzkin*, *K. E. Ralston* and *J. L. Selfridge*, in "The summer meeting in Seattle" by *V. L. …

The MLC conjecture, stating that the mandelbrot set is locally connected.

A stronger version of the MLC conjecture, stating that all multibrots are locally connected. Note that we don't need to require `2 ≤ n` because the conjecture holds in the trivial cases …

All members of the sequence satisfy $n \equiv 108 \pmod{216}$.

Asserts that for any number of colors `r` and any progression length `k`, there always exists some number `N` large enough to guarantee a monochromatic arithmetic progression. In other words, the set …

**Moser's Worm Problem** What is the minimal area (or greatest lower bound on the area) of a shape that can cover every unit-length curve?

There is a set of area 0.260437 that covers all worms. *Reference:* Norwood, Rick; Poole, George (2003), "An improved upper bound for Leo Moser's worm problem", Discrete and Computational Geometry, 29 (3): …

Is there an Euler brick in $n$-dimensional space for any $n > 3$?

For any odd natural number `p` if two of the following conditions hold, then all three must hold: 1. $2^p-1$ is prime 2. $(2^p+1)/3$ is prime 3. Exists a number `k` such …

The **Noether Problem**: let `L` be the field of rational functions in `n` indeterminates over `K`. Is it true that `L/K` has the Noether property? Solution: False.

One can find a counterexample to the Noether Problem's claim by considering a rational function field in 47 indeterminates.

The Noether problem has a positive solution in the four indeterminate case.

The Noether problem has a positive solution in the three indeterminate case.

The Noether problem has a positive solution in the two indeterminate case.

The **no-k-in-line problem**: For $N \geq k$ and $k > 1$, the AllowedSetSize in $(k - 1) * N$, i. e. on an $N \times N$ subset, there is a set of …

In [GK2025] Grebennikov and Kwan prove the no-k-in-line conjecture for $k > 10 ^ 37$.

Noll and Simmons asked, more generally, whether there are solutions to $q_1! \equiv \dots \equiv q_k! \mod p$ for arbitrarily large $k$ (with $q_1 < \dots < q_k$).

For every $n ≥ 3$, there exists $N$ such that any $N$ points, no three on a line, contain a convex $n$-gon.

There are no partition numbers $p(k)$ of the form $x^m$, with $x,m$ integers $>1$. See comment by Zhi-Wei Sun (Dec 02 2013).

Richards [Ri74] showed that only one of the two Hardy-Littlewood conjectures can be true. [Ri74] Richards, Ian (1974). _On the Incompatibility of Two Conjectures Concerning Primes_. Bull. Amer. Math. Soc. 80: 419–438.

For $N \leq 60$, this has been verfied with computers.

**Odd Perfect Number Conjecture.** The Odd Perfect Number Conjecture states that all perfect numbers are even. *Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Perfect_number#Odd_perfect_numbers)

The Odd Perfect Number Conjecture states that all perfect numbers are even.

A known result: If an odd perfect number exists, it must be of the form $p^α * m^2$ where $p$ is prime, $p \equiv 1 \pmod{4}$, $\alpha \equiv 1 \pmod{4}$, and $p …

A known result: If an odd perfect number exists, it must be of the form $p^α * m^2$ where $p$ is prime, $p \equiv 1 \pmod{4}$, $\alpha \equiv 1 \pmod{4}$, and $p …

A known result: If an odd perfect number exists, it must be greater than $10^{1500}$ and must have at least 101 prime factors (including multiplicities). *Reference:* Pascal Ochem, Michaël Rao (2012). "Odd …

A known result: If an odd perfect number exists, it must be greater than $10^{1500}$ and must have at least 101 prime factors (including multiplicities). Reference: Pascal Ochem, Michaël Rao (2012). "Odd …

Conjecture: the dyadic valuation of A93179(n) - 1 does not exceed 2^n - a(n). A93179(n) is minFac(fermatNumber n), the smallest prime factor of the n-th Fermat number. The conjecture states that if …

**Oppermann's Conjecture**: For every integer $x \ge 2$, the following hold: - There exists a prime between $x(x-1)$ and $x^2$. - There exists a prime between $x^2$ and $x(x+1)$.

Ferreira proved that Oppermann's conjecture is true for sufficiently large x.

For every integer $x \ge 2$ there exists a prime between $x(x-1)$ and $x^2$.

For every integer $x \ge 2$ there exists a prime between $x^2$ and $x(x+1)$.

For real number $\alpha$ and prime $p$, $$ \liminf_{n \to\infty} n |n|_{p}\||n\alpha\|| = 0 $$ where $\||x\|| := \min(|x - \lfloor x \rfloor|, |x - \lceil x \rceil|)$ is the distance to …

The pebbling number conjecture: the pebbling number of a Cartesian product of graphs is at most equal to the product of the pebbling numbers of the factors.

Is there a perfect Euler brick?

The Pierce-Birkhoff conjecture states that for every real piecewise-polynomial function `f : ℝⁿ → ℝ`, there exists a finite set of polynomials `gᵢⱼ ∈ ℝ[x₁, ..., xₙ]` such that `f = supᵢ …

The Pierce-Birkhoff conjecture holds for `n = 1`. This was proved by Louis Mahé.

The Pierce-Birkhoff conjecture holds for `n = 2`. This was proved by Louis Mahé.