Graph Theory

Minimal graphs with a prescribed number of spanning trees ★★

Author(s): Azarija; Skrekovski

Conjecture   Let $ n \geq 3 $ be an integer and let $ \alpha(n) $ denote the least integer $ k $ such that there exists a simple graph on $ k $ vertices having precisely $ n $ spanning trees. Then $ \alpha(n) = o(\log{n}). $

Keywords: number of spanning trees, asymptotics

Posted by azi
updated April 22nd, 2012
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Combinatorics

Length of surreal product ★

Author(s): Gonshor

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
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 $$ \displaystyle \sum_{i=1}^{\left \lfloor \frac{\sqrt[3]p}{2} \right \rfloor} r_i = \left \lfloor \frac{\sqrt[3]p}{2} \right \rfloor $$

Keywords: primality

Posted by princeps
updated June 14th, 2013
7 comments
Number Theory

Giuga's Conjecture on Primality ★★

Author(s): Giuseppe Giuga

Conjecture   $ p $ is a prime iff $ ~\displaystyle \sum_{i=1}^{p-1} i^{p-1} \equiv -1 \pmod p $

Keywords: primality

Posted by princeps
updated March 27th, 2012
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