http://openproblemgarden.org/op/alexas_conjecture_on_primality Comments for "Alexa's Conjecture on Primality" en http://openproblemgarden.org/op/alexas_conjecture_on_primality#comment-37644 <p>I've corrected the statement on Alexa's behalf. This version holds up to at least 100 million.</p> Fri, 14 Jun 2013 21:49:24 +0200 Charles R Greathouse IV comment 37644 at http://openproblemgarden.org http://openproblemgarden.org/op/alexas_conjecture_on_primality#comment-7158 <p>your modified version now reduces back to putting mod p on the LHS of the equation, which as I've pointed out above, doesn't work (see counter-examples I gave). Where did you come up with this conjecture? Is there any published reference for it?</p> Mon, 02 Apr 2012 20:19:20 +0200 Anonymous comment 7158 at http://openproblemgarden.org http://openproblemgarden.org/op/alexas_conjecture_on_primality#comment-7157 <p>then please re-word your conjecture, because as it stands, it's ambiguous and not true. It's ambiguous, because the way you defined r_i, one could have chosen r_i, r_i + p, r_i + 2p etc., but when you plug these into the sum, you get a different sum and the equality doesn't make sense.</p> Fri, 30 Mar 2012 23:16:54 +0200 Anonymous comment 7157 at http://openproblemgarden.org http://openproblemgarden.org/op/alexas_conjecture_on_primality#comment-7156 <p>Yes it works .I have checked statement up to 10^6 ,there is no counterexample...</p> Fri, 30 Mar 2012 22:05:39 +0200 princeps comment 7156 at http://openproblemgarden.org http://openproblemgarden.org/op/alexas_conjecture_on_primality#comment-7154 <p>your new statement is ambiguous (which r_i should one choose inside the sum?). I'm assuming you're just trying to move the "mod p" to apply to the sum only (and not to the RHS). If that's what you're doing, it still doesn't work. Same counter-examples at p=66, 102, 246 and 492 for p from 8 to 500.</p> Fri, 30 Mar 2012 19:54:47 +0200 Anonymous comment 7154 at http://openproblemgarden.org http://openproblemgarden.org/op/alexas_conjecture_on_primality#comment-7153 <p>Thanks , I have corrected statement .</p> Fri, 30 Mar 2012 11:26:54 +0200 princeps comment 7153 at http://openproblemgarden.org http://openproblemgarden.org/op/alexas_conjecture_on_primality#comment-7151 <p>formula works for p from 8 through 100, except for p=66.</p> Wed, 28 Mar 2012 22:19:29 +0200 Anonymous comment 7151 at http://openproblemgarden.org http://openproblemgarden.org/op/alexas_conjecture_on_primality <table cellspacing="10"> <tr> <td> Author(s): <a href="/category/alexa">Alexa</a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/category/number_theory_0">Number Theory</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <div class="envsimple"><b>Definition</b>&nbsp;&nbsp; Let <img class="teximage" src="/files/tex/bd013755b6effdbb49a2ee8dcec9023b56000a4a.png" alt="$ r_i $" /> be the unique integer (with respect to a fixed <img class="teximage" src="/files/tex/5214316bdda9be56a163d54dbebb20eeb3f1a0d6.png" alt="$ p\in\mathbb{N} $" />) such that</p> <p><img class="teximage" src="/files/tex/806f13bf880756c911f5da453d0cc2509dabafb3.png" alt="$$(2i+1)^{p-1} \equiv r_i \pmod p ~~\text{ and } ~ 0 \le r_i &lt; p. $$" /> </div> <div class="envtheorem"><b>Conjecture</b>&nbsp;&nbsp; A natural number <img class="teximage" src="/files/tex/2564938a26a5c2d5710bc0f5817232309dc67929.png" alt="$ p \ge 8 $" /> is a prime iff <img class="teximage" src="/files/tex/99af565f4cc4d3bab11eb3fbf54f78626678d484.png" alt="$$ \displaystyle \sum_{i=1}^{\left \lfloor \frac{\sqrt[3]p}{2} \right \rfloor} r_i = \left \lfloor \frac{\sqrt[3]p}{2} \right \rfloor $$" /> </div> </tr></td> </table> </td> </tr> </table> Alexa primality Number Theory http://openproblemgarden.org/op/alexas_conjecture_on_primality#comment Wed, 28 Mar 2012 07:15:06 +0200 princeps 37413 at http://openproblemgarden.org