Unsolved Problems

Conjecture Any linear hypergraph with incidence poset of dimension at most 3 is the intersection hypergraph of a family of triangles and segments in t...

Conjecture Is there a graph $G$ that is a minor-minimal obstruction for two non-orientable surfaces?...

Problem What is the largest connected planar graph of minimum degree 3 which has everywhere positive combinatorial curvature, but is not a prism or an...

Problem Does there exist a finitely presented group of intermediate growth?...

Conjecture Suppose $G$ is a finite group, and $n$ is a positive integer dividing $|G|$. Suppose that $G$ has exactly $n$ solutions to $x^{n} = 1$. Do...

Conjecture If a group has $r$ generators and exponent $n$, is it necessarily finite?...

Conjecture Every finite group is the Galois group of some finite algebraic extension of $\mathbb Q$....

Conjecture There exists a finite lattice that is not an interval in the subgroup lattice of a finite group....

Problem Find a constant $k$ such that for any $d$ there is a sequence of tautologies of depth $k$ that have polynomial (or quasi-polynomial) size proo...

Conjecture Is the theory of the real numbers with the exponential function decidable?...

Question How many steps does it take the sixth Goodstein sequence to terminate?...

Question Can either of the following be expressed in fixed-point logic plus counting: - Given a graph, does it have a perfect matching, i.e., a set $...

Question - Does ${<}\text{-invariant\:FO} = \text{FO}$ hold over graphs of bounded tree-width? - Is ${<}\text{-invariant\:FO}$ included in $\text{MSO...

The problem concerns the extension of Monadic Second Order Logic (over a binary relation representing the edge relation) with the following atomic for...

Let $C$ be a class of finite relational structures. We denote by $f_C(n)$ the number of structures in $C$ over the labeled set $\{0, \dots, n-1 \}$. F...

Question Is the MSO-alternation hierarchy strict for pictures that are balanced, in the sense that the width and the length are polynomially (or linea...

Question Positive Horn logic (pH) is the fragment of FO involving exactly $\exists, \forall, \wedge, =$. Does the fragment $pH \wedge \neg pH$ have th...

Conjecture For every $\varepsilon > 0$ there is $\delta > 0$ such that, for every large $n$, there are $n$-vertex graphs $G$ and $H$ such that $G \equ...

Conjecture Suppose $k$ runners having distinct constant speeds start at a common point and run laps on a circular track with circumference 1. Then for...

Conjecture Every odd prime number must either be adjacent to, or a prime distance away from a primorial or primorial product....

Problem Let $A \subseteq {\mathbb N}$ be a set of $n$ positive integers and set $$ m(A) = - \min_x \sum_{a \in A} \cos(ax). $$ What is $m(n) = \min_A...

Call a polynomial $p \in {\mathbb Q}[x]$ rationally derived if all roots of $p$ and the nonzero derivatives of $p$ are rational. Conjecture There doe...

Conjecture Let $a > b > 0$ be integers. Set $b_1 = b$ and $b_{i+1} = {a \bmod {b_i}}$ for $i \geq 0$. Eventually we have $b_{n+1} = 0$; put $P(a,b) = ...

Conjecture $\pi$ and $e$ are algebraically independent...

Conjecture There is no odd perfect number....

Definition A set of m positive integers $\{a_1, a_2, \dots, a_m\}$ is called a Diophantine $m$-tuple if $a_i\cdot a_j + 1$ is a perfect square for all...

Conjecture There exist infinitely many positive integers $n$ so that both $n$ and $n+2$ are prime....

Conjecture Polignac's Conjecture: For any positive even number n, there are infinitely many prime gaps of size n. In other words: There are infinitely...

Conjecture Let $p$ be a prime natural number. Find all primes $q\equiv1\left(\mathrm{mod}\: p\right)$, such that $2^{\frac{\left(q-1\right)}{p}}\equiv...

Conjecture Find linear transformation construction of primitive pythagorean n-tuple tree!...

Conjecture $3~$ is a primitive root modulo $~p$ for all primes $~p=16\cdot q^4+1$, where $~q>3$ is prime....

Conjecture For all $n > 2$, there exist positive integers $x$, $y$, $z$ such that $$1/x + 1/y + 1/z = 4/n$$....

Conjecture The sequence {L(n) mod m}, where L(n) are the Lucas numbers, contains a complete residue system modulo m if and only if m is one of the fol...

Conjecture Every even number greater than $10$ can be represented as the sum of an odd prime number and an odd semiprime....

Conjecture $p$ is a prime iff $~\displaystyle \sum_{i=1}^{p-1} i^{p-1} \equiv -1 \pmod p$...

Definition Let $r_i$ be the unique integer (with respect to a fixed $p\in\mathbb{N}$ ) such that $$(2i+1)^{p-1} \equiv r_i \pmod p ~~\text{ and } ~ 0...

Conjecture Let $E/K$ be an elliptic curve over a number field $K$. Then the order of the zeros of its $L$-function, $L(E, s)$, at $s = 1$ is the Morde...

Let $B \subseteq {\mathbb N}$. The representation function $r_B: {\mathbb N} \rightarrow {\mathbb N}$ for $B$ is given by the rule $r_B(k) = \#\{ (i,j...

Conjecture Every even integer greater than 2 is the sum of two primes....

Conjecture If every second positive integer except 2 is remaining, then every third remaining integer except 3, then every fourth remaining integer et...

The zeroes of the Riemann zeta function that are inside the Critical Strip (i.e. the vertical strip of the complex plane where the real part of the co...

Conjecture Given any $n$ complex numbers $z_1,...,z_n$ which are linearly independent over the rational numbers $\mathbb{Q}$, then the extension field...

Problem Let the notation $a|b$ denote " $a$ divides $b$ ". The mimic function in number theory is defined as follows [1]. Definition For any positiv...

Conjecture For any $\epsilon>0$ $$\zeta\left(\frac12 + it\right) \mbox{ is }\mathcal{O}(t^\epsilon).$$ Since $\epsilon$ can be replaced by a smaller ...

Question Is Euler-Mascheroni constant an transcendental number?...

Conjecture Skewes' number $e^{e^{e^{79}}}$ is not an integer....

Conjecture Are all Fermat Numbers $$ F_n = 2^{2^{n } } + 1 $$ Square-Free?...

Conjecture A Fermat prime is a Fermat number $$ F_n = 2^{2^n } + 1 $$ that is prime. The only known Fermat primes are F_0 =3,F_1=5,F_2=17,F_3 =257,F_...

Conjecture Are all Mersenne Numbers with prime exponent ${2^p-1}$ Square free?...

Conjecture A Mersenne prime is a Mersenne number $$ M_n = 2^p - 1 $$ that is prime. Are there infinite number of Mersenne Primes?...