Open Problem Garden

Conjecture Every surreal number has a unique sign expansion, i.e. function $f: o\rightarrow \{-, +\}$ , where $o$ is some ordinal. This $o$ is the length of given sign expansion and also the birthday of the corresponding surreal number. Let us denote this length of $s$ as $\ell(s)$ .

It is easy to prove that

$\ell(s+t) \leq \ell(s)+\ell(t)$

What about

$\ell(s\times t) \leq \ell(s)\times\ell(t)$

Keywords: surreal numbers

Posted by Lukáš Lánský
updated June 5th, 2012

2 comments

Alexa

Number Theory

Alexa's Conjecture on Primality ★★

Author(s): Alexa

Definition Let $r_i$ be the unique integer (with respect to a fixed $p\in\mathbb{N}$ ) such that

$(2i+1)^{p-1} \equiv r_i \pmod p ~~\text{ and } ~ 0 \le r_i < p.$

Conjecture A natural number $p \ge 8$ is a prime iff $\displaystyle \sum_{i=1}^{\left \lfloor \frac{\sqrt[3]p}{2} \right \rfloor} r_i = \left \lfloor \frac{\sqrt[3]p}{2} \right \rfloor$