Browse Problems

The **Köthe conjecture**: In any ring, the sum of two nil left ideals is nil.

The **Köthe conjecture**: for any nil ideal `I` of `R`, the matrix ideal `M_n(I)` is a nil ideal of the matrix ring `M_n(R)`.

The **Köthe conjecture**: every left nil radical is contained in the Köthe radical.

The **Köthe conjecture**: for any positive integer `n`, the Köthe radical of `R` is the matrix ideal `M_2(Nil*(R))`.

The **Köthe conjecture**: for any nil ideal `I` of `R`, the matrix ideal `M_2(I)` is a nil ideal of the matrix ring `M_2(R)`.

Let $G$ and $H$ be finite groups of the same order with $\sum_{g \in G} \phi(|g|) = \sum_{h \in H} \phi(|h|)$, where $\phi$ is the Euler totient function. Suppose that $G$ is …

Let $G$ be a finite $p$-group and assume that all abelian normal subgroups of $G$ have order at most $p^k$. Is it true that every abelian subgroup of $G$ has order at …

Krückeberg ([Kr61]) exhibited an infinite Sidon set `A` with `sidonUpperDensity A = 1 / Real.sqrt 2`, improving Erdős’ earlier `1 / 2` lower bound. [Kr61] Krückeberg, Fritz, $B\sb{2}$-Folgen und verwandte Zahlenfolgen. J. …

Kummer–Vandiver conjecture states that for every prime $p$, the class number of the maximal real subfield of $\mathbb{Q}(\zeta_p)$ is not divisible by $p$. -

## Kurepa's conjecture For all $n$, $$!n\not\equiv 0 \mod n$$ This appears as B44 "Sums of factorials." in [Unsolved Problems in Number Theory](https://doi.org/10.1007/978-0-387-26677-0) by *Richard K. Guy*

An equivalent formulation in terms of the gcd of $n!$ and $!n$.

This statement can be reduced to the prime case only.

In the case of large n, the problem was solved in [On the largest product-free subsets of the alternating groups](https://arxiv.org/pdf/2205.15191). Specifically, this theorem formalizes the statement of theorem 1.1 in the mentioned …

The Latin Tableau Conjecture: If G is the simple graph of a Young diagram, then G is CDS-colorable.

What is the smallest square that can contain 11 unit squares? Reference: [Wikipedia](https://en.wikipedia.org/wiki/Square_packing#In_a_square)

What is the smallest circle that can contain 15 unit circles? Reference: Graham RL, Lubachevsky BD, Nurmela KJ, Ostergard PRJ. Dense packings of congruent circles in a circle. Discrete Math 1998;181:139–154.

What is the smallest square that can contain 17 unit squares? Reference: [Wikipedia](https://en.wikipedia.org/wiki/Square_packing#In_a_square)

What is the smallest circle that can contain 3 unit squares? Reference: [Wikipedia](https://en.wikipedia.org/wiki/Square_packing#In_a_circle)

What is the smallest square that can contain 21 unit circles?

**Lebesgue-Nagell Equation Conjecture** For any odd prime $p$, the only integer solutions $(x, y)$ to the equation $x^2 - 2 = y^p$ are $(x, y) = (\pm 1, -1)$. *Reference:* Ethan Katz …

Does there always exist at least one prime between consecutive perfect squares?

Ferreira proved that the conjecture is true for sufficiently large n.

Let `M(f)` denote the Mahler measure of `f`. There exists a constant `μ>1` such that for any `f(x)∈ℤ[x], M(f)>1 → M(f)≥μ`.

`μ=M(X^10 + X^9 - X^7 - X^6 - X^5 - X^4 - X^3 + X + 1)` is the best value for `lehmer_mahler_measure_problem`.

If $f$ is not reciprocal and $M(f) > 1$ then $M(f) \ge M(X^3 - X - 1)$.

If all the coefficients of $f$ are odd and $M(f) > 1$, then $M(f) \ge M(X^2 - X - 1)$.

Does there exist a composite number $n > 1$ such that Euler’s totient function $\varphi(n)$ divides $n - 1$?

For all odd integers $n ≥ 7$ there are prime numbers $p,q$ such that $n = p+2q$.

For all odd integers $n ≥ 9$ there are odd prime numbers $p,q,r,s$ and natural numbers $a,b$ such that $p+2q = n$, $2+pq = 2^a+r$, $2p+q = 2^b+s$

For any two real numbers $\alpha$ and $\beta$, $$ \liminf_{n\to\infty} n\||n\alpha\||\||n\beta\|| = 0 $$ where $\||x\|| := \min(|x - \lfloor x \rfloor|, |x - \lceil x \rceil|)$ is the distance to the …

Consider $n$ runners on a circular track of unit length. At the initial time $t = 0$, all runners are at the same position and start to run; the runners' speeds are …

Asymptotic lower bound (1.2) in [Aa19]. Named after Hardy and Littlewood [HaL28].

The **Mahler Conjecture** states that there are no non-zero Z-numbers.

It is known that for all rational `p/q > 1` in lowest terms, we have `Ω(p/q) > 1/p`.

Does the 6-sphere admit a complex structure, i.e. an atlas of holomorphically compatible charts relating it to `EuclideanSpace ℂ (Fin 3)`?

Is there any polynomial $f(x, y) \in \mathbb{Q}[x, y]$ such that $f : \mathbb{Q} \times \mathbb{Q} \rightarrow \mathbb{Q}$ is a bijection?

Does there exist a bijection $f : ℝ^n → ℝ^n$ such that $f$ is connected but the inverse is not?

There exists a connected bijection ℝ → ℝ^2 where the inverse is not connected, proven in [mathoverflow/260589](https://mathoverflow.net/questions/260589) by user [Gro-Tsen](https://mathoverflow.net/users/17064/gro-tsen).

Does there exist a category that is pretriangulated but not triangulated?

Let $P(x), Q(x) ∈ ℝ[x]$ be two monic polynomials with non-negative coefficients. If $R(x) = P(x)Q(x)$ is a $0,1$ polynomial (coefficients only from $\{0,1\}$), then $P(x)$ and $Q(x)$ are also $0, 1$ …

Let $f : \mathbb R^n \to \mathbb R, n \geq 2$ be a $C^1$ function. Is it true that $$\sup_{x \in \mathbb R^n}f(x) = \sup_{x\in \mathbb R^n} f(x+\nabla f(x))$$?

Let $f : \mathbb R^n \to \mathbb R, n \geq 2$ be a $C^1$ function. Is the boundedness of $\sup_{x \in \mathbb R^n}f(x)$ and $\sup_{x\in \mathbb R^n} f(x+\nabla f(x))$ equivalent?

Let $f : \mathbb R^n \to \mathbb R, n \geq 2$ be a $C^1$ function. Does the equality $$\sup_{x \in \mathbb R^n}f(x) = \sup_{x\in \mathbb R^n} f(x+\nabla f(x))$$ hold when both suprema …

Moreover, whenever $n$ is a perfect square we have $F(n) \geq n^{3/2}$.

For all $n$ we have $F(n) \leq n^2$.

$F(n) \leq n^2 / \exp(\Omega(\log^*(n)))$.

$F(n) \leq n^{3/2}$.

Suppose that for some $n$ we have $F(n) = n ^ {\alpha}$. Then there are arbitrarily large $m$ such that $F(m) \geq m^{\alpha}$.

Given a complex polynomial $p$ of degree $d ≥ 2$ and a complex number $z$ there is a critical point $c$ of $p$, such that $|p(z)-p(c)|/|z-c| ≤ |p'(z)|$.