Are all Mersenne Numbers with prime exponent square-free? ★★★


\begin{conjecture} Are all Mersenne Numbers with prime exponent ${2^p-1}$ Square free? \end{conjecture}

Keywords: Mersenne number

What are hyperfuncoids isomorphic to? ★★

Author(s): Porton

Let $\mathfrak{A}$ be an indexed family of sets.

\emph{Products} are $\prod A$ for $A \in \prod \mathfrak{A}$.

\emph{Hyperfuncoids} are filters $\mathfrak{F} \Gamma$ on the lattice $\Gamma$ of all finite unions of products.

\begin{problem} Is $\bigcap^{\mathsf{\tmop{FCD}}}$ a bijection from hyperfuncoids $\mathfrak{F} \Gamma$ to: \begin{enumerate} \item prestaroids on $\mathfrak{A}$; \item staroids on $\mathfrak{A}$; \item completary staroids on $\mathfrak{A}$? \end{enumerate} If yes, is $\operatorname{up}^{\Gamma}$ defining the inverse bijection? If not, characterize the image of the function $\bigcap^{\mathsf{\tmop{FCD}}}$ defined on $\mathfrak{F} \Gamma$.

Consider also the variant of this problem with the set $\Gamma$ replaced with the set $\Gamma^{\ast}$ of complements of elements of the set $\Gamma$. \end{problem}

Keywords: hyperfuncoids; multidimensional

Domain and image for Gamma-reloid ★★

Author(s): Porton

\begin{conjecture} $\ensuremath{\operatorname{dom}}( \mathsf{\ensuremath{\operatorname{RLD}}})_{\ensuremath{\operatorname{\Gamma}}} f =\ensuremath{\operatorname{dom}}f$ and $\ensuremath{\operatorname{im}}( \mathsf{\ensuremath{\operatorname{RLD}}})_{\ensuremath{\operatorname{\Gamma}}} f =\ensuremath{\operatorname{im}}f$ for every funcoid $f$. \end{conjecture}


Another conjecture about reloids and funcoids ★★

Author(s): Porton

\begin{definition} $\square f = \bigcap^{\mathsf{RLD}} \mathrm{up}^{\Gamma (\operatorname{Src} f ; \operatorname{Dst} f)} f$ for reloid $f$. \end{definition}

\begin{conjecture} $(\mathsf{RLD})_{\Gamma} f = \square (\mathsf{RLD})_{\mathrm{in}} f$ for every funcoid $f$. \end{conjecture}

Note: it is known that $(\mathsf{RLD})_{\Gamma} f \ne \square (\mathsf{RLD})_{\mathrm{out}} f$ (see below mentioned online article).


One-way functions exist ★★★★


\begin{conjecture} \Def[One-way functions]{One-way_function} exist. \end{conjecture}

Keywords: one way function

Funcoid corresponding to reloid through lattice Gamma ★★

Author(s): Porton

\begin{conjecture} For every reloid $f$ and $\mathcal{X} \in \mathfrak{F} (\operatorname{Src} f)$, $\mathcal{Y} \in \mathfrak{F} (\operatorname{Dst} f)$: \begin{enumerate} \item $\mathcal{X} \mathrel{[(\mathsf{FCD}) f]} \mathcal{Y} \Leftrightarrow \forall F \in \operatorname{up}^{\Gamma (\operatorname{Src} f ; \operatorname{Dst} f)} f : \mathcal{X} \mathrel{[F]} \mathcal{Y}$; \item $\langle (\mathsf{FCD}) f \rangle \mathcal{X} = \bigcap_{F \in \operatorname{up}^{\Gamma (\operatorname{Src} f ; \operatorname{Dst} f)} f} \langle F \rangle \mathcal{X}$. \end{enumerate} \end{conjecture}

It's proved by me in \href [this online article]{}.

Keywords: funcoid corresponding to reloid