http://openproblemgarden.org/op/special_m Comments for "Special M" en http://openproblemgarden.org/op/special_m <table cellspacing="10"> <tr> <td> Author(s): <a href="/category/kimberling_clark">Kimberling</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> <p>Let <img class="teximage" src="/files/tex/535dee6c3b72bcc4d571239ed00be162ee1e6fbe.png" alt="$ r $" /> denote the golden ratio, <img class="teximage" src="/files/tex/7a485af3aeee08213a103eb0625aedf1b37957d6.png" alt="$ \frac{(1+\sqrt{5})}{2} $" /> and let <img class="teximage" src="/files/tex/9d729d162d574175df1658e073a33e85bf61dd6e.png" alt="$ \lfloor \rfloor $" /> denote the floor function. For fixed <img class="teximage" src="/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png" alt="$ n $" />, let <img class="teximage" src="/files/tex/0378d185b4bc778831affeb7488d83abd2c03f1c.png" alt="$ u(k) = \lfloor kr^n \rfloor $" />, let <img class="teximage" src="/files/tex/75b3b41beb34e06ed47edb024f9dac67d538f021.png" alt="$ v(k) = \lfloor kr \rfloor^n $" />, and let <img class="teximage" src="/files/tex/3219e7cbe7818e865ac97d78f6f8a0b417449150.png" alt="$ w(k) = \left \lfloor \frac{v(k)}{k^{(n-1)}} \right \rfloor $" />. We can expect <img class="teximage" src="/files/tex/f41b7e069371ec2311409f36752c88d66e9bf8fc.png" alt="$ w $" /> to have about the same growth rate as <img class="teximage" src="/files/tex/06183efdad837019eb0937c4e40f9e7beaa2e8d8.png" alt="$ u $" />.</p> <div class="envtheorem"><b>Conjecture</b>&nbsp;&nbsp; Prove or disprove that for every fixed <img class="teximage" src="/files/tex/b07c8b96507c45835ae7d500e2639d1c292a01fa.png" alt="$ n &gt; 0 $" />, as <img class="teximage" src="/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png" alt="$ k $" /> ranges through all the positive integers, there is a number <img class="teximage" src="/files/tex/3f02401f624e31ef8679d3c3628c1f310058f388.png" alt="$ M $" /> such that <img class="teximage" src="/files/tex/db5855b5c63d42bcd06beacb7a167baea8a859c3.png" alt="$ u(k) - w(k) $" /> takes each of the values <img class="teximage" src="/files/tex/59b85d9e57d6e576ec493cace882eb416e7a7d8a.png" alt="$ 1,2,\dots,M $" /> infinitely many times, and <img class="teximage" src="/files/tex/3ba59e71469b38952b2792dcf7365dc7e154625e.png" alt="$ u(k) - w(k) \leq M $" />. (Can you formulate <img class="teximage" src="/files/tex/3f02401f624e31ef8679d3c3628c1f310058f388.png" alt="$ M $" /> as a function of <img class="teximage" src="/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png" alt="$ n $" />? Generalize for other numbers <img class="teximage" src="/files/tex/535dee6c3b72bcc4d571239ed00be162ee1e6fbe.png" alt="$ r $" />?) </div> </tr></td> </table> </td> </tr> </table> Kimberling, Clark Number Theory http://openproblemgarden.org/op/special_m#comment Tue, 12 Jun 2012 15:08:25 +0200 vprusso 37466 at http://openproblemgarden.org