Unsolved Problems

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

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) = ...

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

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

Conjecture If $a_1,a_2,\ldots,a_{2n-1}$ is a sequence of elements from a multiplicative group of order $n$, then there exist $1 \le j_1 < j_2 \ldots <...

Conjecture For every $0 \le t \le n-1$, the sequence in ${\mathbb Z}_n^2$ consisting of $n-1$ copes of $(1,0)$ and $t$ copies of $(0,1)$ has the fewes...

For every finite multiplicative group $G$, let $s(G)$ ( $s'(G)$ ) denote the smallest integer $m$ so that every sequence of $m$ elements of $G$ has a ...

Problem Does for every integer $N$ exist a covering system with all moduli distinct and at least equal to~ $N$?...

Problem (1) Prove that there exist infinitely many positive integers $n$ such that $$\gcd({2n\choose n}, 3\cdot 5\cdot 7) = 1.$$ Problem (2) Prove th...

Problem Find an explicit formula for Frobenius number $g(a_1, a_2, \dots, a_n)$ of co-prime positive integers $a_1, a_2, \dots, a_n$ for $n\geq 4$....

Conjecture There is a finite upper bound on the multiplicities of entries in Pascal's triangle, other than the number $1$. The number $2$ appears onc...

Problem Find six positive integers $x_1, x_2, \dots, x_6$ such that $$x_1^6 + x_2^6 + x_3^6 + x_4^6 + x_5^6 = x_6^6$$ or prove that such integers do n...

Problem (1) Find a counterexample to Baillie-PSW primality test or prove that there is no one. Problem (2) Find a composite $n\equiv 3$ or $7\pmod{10...

Conjecture For any prime $p$, there exists a Fibonacci number divisible by $p$ exactly once. Equivalently: Conjecture For any prime $p>5$, $p^2$ doe...

Question Does there exist a $3\times 3$ magic square composed of distinct perfect squares?...

Conjecture Does a perfect cuboid exist?...

Conjecture If $X_1, \dotsc, X_n \geq 0$ are independent random variables with $\mathbb{E}[X_i] \leq \mu$, then $$\mathrm{Pr} \left( \sum X_i - \mathbb...

Conjecture The famous 0-1 Knapsack problem is: Given $a_{1},a_{2},\dots,a_{n}$ and $b$ integers, determine whether or not there are $0-1$ values $x_{...

Question What is the complexity of the following problem? Given $a_1,\dots,a_n; k$, determine whether or not $\sum_i \sqrt{a_i} \leq k.$...

Problem Can $O(n)$-size circuits compute the function $f$ on $\{0,1,2\}^*$ defined inductively by $f(\lambda) = \lambda$, $f(0x) = 0f(x)$, $f(1x) = 1f...

Question $S(S(f)) = S(f)$ for every endo-reloid $f$?...

Conjecture Does every Jordan curve have 4 points on it which form the vertices of a square?...

Problem Given a link $L$ in $S^3$, let the symmetry group of $L$ be denoted $Sym(L) = \pi_0 Diff(S^3,L)$ ie: isotopy classes of diffeomorphisms of $S^...

Problem Does there exist a subset of $\mathbb R^3$ such that its fundamental group has an element of finite order?...

Problem Ian Agol and Matthias Goerner observed that the 4x5 chessboard complex is the complement of many distinct links in the 3-sphere. Their observa...

Question $( \mathsf{RLD})_{\mathrm{out}} (f \cap^{\mathsf{FCD}} ( \mathcal{A} \times^{\mathsf{FCD}} \mathcal{B})) = (( \mathsf{RLD})_{\mathrm{out}} f)...

Conjecture Let $C$ be a Cantor set embedded in $\mathbb{R}^n$. Is there a self-homeomorphism $f$ of $\mathbb{R}^n$ for every $\epsilon$ greater than $...

Conjecture Does any path-connected, compact set in the plane which does not separate the plane have the fixed point property? A set has the fixed poi...

Conjecture Let $G$ be a locally compact topological group. If $G$ has a continuous faithful group action on an $n$-manifold, then $G$ is a Lie group....

Conjecture $\mathcal{A} \times^{\mathsf{\ensuremath{\operatorname{RLD}}}}_F \mathcal{B} \subset \mathcal{A} \ltimes \mathcal{B} \subset \mathcal{A} \t...

Conjecture (solved) If $a \times^{\mathsf{\ensuremath{\operatorname{RLD}}}} b \subseteq \left( \mathsf{\ensuremath{\operatorname{RLD}}} \right)_{\ensu...

Let $\mho$ be a set, $\mathfrak{F}$ be the set of filters on $\mho$ ordered reverse to set-theoretic inclusion, $\mathfrak{P}$ be the set of principal...