Browse Problems

**Artin's Conjecture on Primitive Roots**, second half, conditional on GRH.

**Artin's Conjecture on Primitive Roots**, second half, power version, conditional on GRH.

**Artin's Conjecture on Primitive Roots**, second half, power version, conditional on GRH

Conjecture p.579 in [Aa19]: $\left({1}{3} + o(1)\right) n^2$.

**Zhi-Wei Sun's 1680-Conjecture (A280831)**: Any nonnegative integer can be written as $x^2 + y^2 + z^2 + w^2$ with $x, y, z, w$ nonnegative integers such that $x^4 + 1680 y^3 z$ …

**Zhi-Wei Sun's Conjecture (A239957)**: Every prime $p$ has a primitive root $0 < g < p$ of the form $k^2 + 1$, where $k$ is an integer.

**Zhi-Wei Sun's Four-Square Conjecture (A308734)**: Any integer $n > 1$ can be written as $(2^a \cdot 3^b)^2 + (2^c \cdot 5^d)^2 + x^2 + y^2$ for nonnegative integers $a, b, c, d, …

**Zhi-Wei Sun's 2-4-6-8 Conjecture (A306477)**: Any integer $n > 0$ can be written as $\binom{w+2}{2} + \binom{x+3}{4} + \binom{y+5}{6} + \binom{z+7}{8}$ for nonnegative integers $w, x, y, z$.

**Zhi-Wei Sun's Conjecture (A287616)**: Any nonnegative integer can be written as the sum of a triangular number $x(x+1)/2$, a generalized pentagonal number $y(3y+1)/2$, and a generalized heptagonal number $z(5z+1)/2$, where $x, y, …

**Conjecture (A6697)**: The generating function for the number of subwords of length $n$ in the infinite word generated by $a \mapsto aab, b \mapsto b$ is $$\sum_{n \geq 0} a_n x^n = …

**Zhi-Wei Sun's Conjecture (A232174)**: Any integer $n > 1$ can be written as $x + y$ with $x, y > 0$ such that both $x + ny$ and $x^2 + ny^2$ are …

**Zhi-Wei Sun's Conjecture (A303656)**: Any integer $n > 1$ can be written as the sum of two squares, a power of 3, and a power of 5.

**Zhi-Wei Sun's Conjecture (A281976)**: Any integer $n \geq 0$ can be written as $x^2 + y^2 + z^2 + w^2$ with $x, y, z, w$ nonnegative integers and $z \leq w$, such …

WOWII [Conjecture 1](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/) For a simple connected graph `G` the maximum number of leaves of a spanning tree satisfies `Ls(G) ≥ n(G) + 1 - 2·m(G)` where `n(G)` counts vertices and `m(G)` …

**Conjecture 1.1**: For any odd prime $k$, the sum associated with the classical theta function $θ_3$, $S(k)$ is positive.

Does every finite partially ordered set that is not totally ordered contain two elements $x$ and $y$ such that the probability that $x$ appears before $y$ in a random linear extension is …

WOWII [Conjecture 19](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/) If `G` is connected then the size `b(G)` of a largest induced bipartite subgraph satisfies `b(G) ≥ FLOOR((∑ ecc(v))/(|V|) + sSup (range (l G)))`, where `ecc(v)` denotes eccentricity and …

WOWII [Conjecture 2](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/) For a simple connected graph `G`, `Ls(G) ≥ 2 · (l(G) - 1)` where `l(G)` is the average independence number of the neighbourhoods of the vertices of `G`.

WOWII [Conjecture 3](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/) For a connected simple graph `G`, the number of leaves in a maximum spanning tree satisfies `Ls(G) ≥ gi(G) * MaxTemp(G)`, where `gi(G)` is the independent domination number and …

*Conjecture 3.4* There exists $0 < \delta < 1$ such that for any $a \in \mathbb{S}^3$, there exists $b \in \mathbb{Z}^4$ and $k \in \mathbb{Z}$ such that $\|b\| = 5^k$ and $\langle …

WOWII [Conjecture 34](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/) Let `path(G)` be the floor of the average distance of a connected graph `G`. Then `path(G) ≥ ceil( distavg(G, center) + distavg(G, maxEccentricityVertices G) )`.

WOWII [Conjecture 4](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/) If `G` is a connected graph then the maximum number of leaves over all spanning trees satisfies `Ls(G) ≥ NG(G) - 1` where `NG(G)` is the minimal neighbourhood size …

WOWII [Conjecture 40](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/) For a nontrivial connected graph `G` the size `f(G)` of a largest induced forest satisfies `f(G) ≥ ceil((p(G) + b(G) + 1)/2)` where `p(G)` is the path cover number …

**Conjecture 4.1**: For any prime $k$ larger than $5$, $S(k) > k$.

**Conjecture 4.2**: For any prime $k$ larger than $233$, $S(k) > 2k$.

**Conjecture 4.3**: For any prime $k$ larger than $3119$, $S(k) > 3k$.

**Conjecture 4.4**: Given a natural number $n ∈ ℕ$, for all large enough odd prime $k$ (depending on $n$), $nk < S(k)$.

WOWII [Conjecture 5](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/) For a simple connected graph `G`, `Ls(G)` is bounded below by the maximal size of a sphere of radius `radius(G)` around the centres of `G`.

WOWII [Conjecture 58](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/) For a connected graph `G`, the size `f(G)` of a largest induced forest satisfies `f(G) ≥ ceil( b(G) / average l(v) )` where `b(G)` is the largest induced bipartite …

WOWII [Conjecture 6](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/) For a connected graph `G` we have `Ls(G) ≥ 1 + n(G) - m(G) - a(G)` where `a(G)` is defined via independent sets.

**Conjecture (A81091)**: There are infinite primes of the form $2^n + 2^i + 1$, with $0 < i < n$.

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

The minimal area of a convex shape that can cover every unit-length curve is attained. This follows from the Blaschke selection theorem.

0.232239 is a lower bound on the area of a convex set that covers all worms. *Reference:* Khandhawit, Tirasan; Pagonakis, Dimitrios; Sriswasdi, Sira (2013), "Lower Bound for Convex Hull Area and Universal …

There is a convex set of area 0.270911861 that covers all worms. *Reference:* Wang, Wei (2006), "An improved upper bound for the worm problem", Acta Mathematica Sinica, 49 (4): 835–846, MR 2264090.

Does there exist an undirected graph with 99 vertices, in which each two adjacent vertices have exactly one common neighbor, and in which each two non-adjacent vertices have exactly two common neighbors? …

Problem 16 in [Ar2013]: Is the cardinality of every homogeneous ω-monolithic compact space not greater than 𝔠?

Problem 17 in [Ar2013]: Is it true that every ω-monolithic compact space contains a point with a first countable neighborhood basis?

The **Unit Conjecture** is false. At least there is a counterexample for any prime and zero characteristic: [Mu21] Murray, A. (2021). More Counterexamples to the Unit Conjecture for Group Rings. [Pa21] Passman, …

There is a counterexample to **Unit Conjecture** in any characteristic.

If $n = ab$ is a crystal, then there are no other pairs of positive integers $c, d > 1$, different from the couple $a, b$, such that $n = cd$ and …

The first Cuboid conjecture

In [Sh12], Ruslan notes that a perfect Euler brick does not exist if all three Cuboid conjectures hold.

The third Cuboid conjecture

The second Cuboid conjecture

The sequence will eventually reach $1$.

Danilov proved that one cannot replace the exponent $1/2$ with larger number. In other words, for any $\delta > 0$, there is no positive constant $C$ such that $|y^2 - x^3| > …

For each $m$ and $\alpha$, the density of the set of integers which are divisible by some $d \equiv 1 \pmod{m}$ with $1 < d < \exp (m ^ \alpha)$ exists.