Importance: Medium ✭✭
Author(s): Alexa
Subject: Number Theory
Keywords: primality
Recomm. for undergrads: no
Posted by: princeps
on: March 28th, 2012
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 $$ \displaystyle \sum_{i=1}^{\left \lfloor \frac{\sqrt[3]p}{2} \right \rfloor} r_i = \left \lfloor \frac{\sqrt[3]p}{2} \right \rfloor $$

The conjecture is obviously true when $ p $ is prime, so it suffices to check when $ p $ is composite.


* indicates original appearance(s) of problem.


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