Unsolved Problems

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...

Conjecture The poset of completary multifuncoids of the form $(\mathscr{P}\mho)^n$ is for every sets $\mho$ and $n$: - atomic; - atomistic. See belo...

Conjecture The poset of multifuncoids of the form $(\mathscr{P}\mho)^n$ is for every sets $\mho$ and $n$: - atomic; - atomistic. See below for defin...

Conjecture Let $F$ is a family of multifuncoids such that each $F_i$ is of the form $\lambda j \in N \left( i \right): \mathfrak{F} \left( U_j \right)...

Conjecture Let $f_1$ and $f_2$ are monovalued, entirely defined funcoids with $\operatorname{Src}f_1=\operatorname{Src}f_2=A$. Then there exists a poi...

Conjecture For composable reloids $f$ and $g$ it holds - $\operatorname{Compl} ( g \circ f) = ( \operatorname{Compl} g) \circ f$ if $f$ is a co-compl...

Conjecture Every metamonovalued funcoid is monovalued. The reverse is almost trivial: Every monovalued funcoid is metamonovalued....

Conjecture Every metamonovalued reloid is monovalued....

Let $\delta$ be a proximity. A set $A$ is connected regarding $\delta$ iff $\forall X,Y \in \mathscr{P} A \setminus \{ \emptyset \}: \left( X \cup Y ...

Conjecture Let $f$ is a $T_1$-separable (the same as $T_2$ for symmetric transitive) compact funcoid and $g$ is a uniform space (reflexive, symmetric,...

Definition $\square f = \bigcap^{\mathsf{RLD}} \mathrm{up}^{\Gamma (\operatorname{Src} f; \operatorname{Dst} f)} f$ for reloid $f$. Conjecture $(\mat...

Let $\mathfrak{A}$ be an indexed family of sets. Products are $\prod A$ for $A \in \prod \mathfrak{A}$. Hyperfuncoids are filters $\mathfrak{F} \Gam...

Conjecture $f \sqcap \bigsqcup S = \bigsqcup \langle f \sqcap \rangle^{\ast} S$ for principal funcoid $f$ and a set $S$ of funcoids of appropriate sou...