# primality

## Alexa's Conjecture on Primality ★★

Author(s): Alexa

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

$$(2i+1)^{p-1} \equiv r_i \pmod p ~~\text{ and } ~ 0 \le r_i < p.$$ \end{definition} \begin{conjecture} A natural number $p \ge 8$ is a prime iff $$\displaystyle \sum_{i=1}^{\left \lfloor \frac{\sqrt[3]p}{2} \right \rfloor} r_i = \left \lfloor \frac{\sqrt[3]p}{2} \right \rfloor$$ \end{conjecture}

Keywords: primality

## Giuga's Conjecture on Primality ★★

Author(s): Giuseppe Giuga

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

Keywords: primality