Open Problem Garden

Author(s):

Conjecture A Mersenne prime is a Mersenne number $M_n = 2^p - 1$ that is prime.

Are there infinite number of Mersenne Primes?

Keywords: Mersenne number; Mersenne prime

Posted by kurtulmehtap
updated February 4th, 2015

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Number Theory » Analytic N.T.

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

Author(s):

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

Keywords: Mersenne number

Posted by kurtulmehtap
updated February 4th, 2015

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What are hyperfuncoids isomorphic to? ★★

Author(s): Porton

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

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

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

Problem Is $\bigcap^{\mathsf{\tmop{FCD}}}$ a bijection from hyperfuncoids $\mathfrak{F} \Gamma$ to:

$\mathfrak{A}$

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 $\Gamma$ .

Keywords: hyperfuncoids; multidimensional

Posted by porton
updated December 9th, 2014

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Domain and image for Gamma-reloid ★★

Author(s): Porton

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 $f$ .