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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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Number Theory

MacEachen Conjecture ★

Author(s): McEachen

Conjecture   Every odd prime number must either be adjacent to, or a prime distance away from a primorial or primorial product.

Keywords: primality; prime distribution

Posted by billymac00
updated July 7th, 2012
4 comments | 1 attachment
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