Open Problem Garden

Lucas Numbers Modulo m ★★

Author(s):

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 following: 2, 4, 6, 7, 14, 3^k, k >=1.

Keywords: Lucas numbers

Posted by Martin Erickson
updated July 1st, 2014

1 comment

Erdős–Straus conjecture ★★

Author(s): Erdos; Straus

Conjecture

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

Keywords: Egyptian fraction

Posted by ACW
updated July 14th, 2014

4 comments

Number Theory

3 is a primitive root modulo primes of the form 16 q^4 + 1, where q>3 is prime ★★

Author(s):

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

Keywords:

Posted by princeps
updated August 24th, 2012

1 comment

Topology

Graph product of multifuncoids ★★

Author(s): Porton

Conjecture Let $F$ is a family of multifuncoids such that each $F_i$ is of the form $\lambda j \in N \left( i \right) : \mathfrak{F} \left( U_j \right)$ where $N \left( i \right)$ is an index set for every $i$ and $U_j$ is a set for every $j$ . Let every $F_i = E^{\ast} f_i$ for some multifuncoid $f_i$ of the form $\lambda j \in N \left( i \right) : \mathfrak{P} \left( U_j \right)$ regarding the filtrator $\left( \prod_{j \in N \left( i \right)} \mathfrak{F} \left( U_j \right) ; \prod_{j \in N \left( i \right)} \mathfrak{P} \left( U_j \right) \right)$ . Let $H$ is a graph-composition of $F$ (regarding some partition $G$ and external set $Z$ ). Then there exist a multifuncoid $h$ of the form $\lambda j \in Z : \mathfrak{P} \left( U_j \right)$ such that $H = E^{\ast} h$ regarding the filtrator $\left( \prod_{j \in Z} \mathfrak{F} \left( U_j \right) ; \prod_{j \in Z} \mathfrak{P} \left( U_j \right) \right)$ .