http://openproblemgarden.org/op/singmasters_conjecture Comments for "Singmaster's conjecture" en http://openproblemgarden.org/op/singmasters_conjecture <table cellspacing="10"> <tr> <td> Author(s): <a href="/category/singmaster_david">Singmaster</a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/category/number_theory_0">Number Theory</a> » <a href="/category/combinatorial_number_theory">Combinatorial N.T.</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <div class="envtheorem"><b>Conjecture</b>&nbsp;&nbsp; There is a finite upper bound on the multiplicities of entries in Pascal's triangle, other than the number <img class="teximage" src="/files/tex/f81bea742c6e484717a25f7b16835462361c1d2e.png" alt="$ 1 $" />. </div> <p> The number <img class="teximage" src="/files/tex/5271e36bb1c040e0f14061d89cd97d0c86d4e06f.png" alt="$ 2 $" /> appears once in Pascal's triangle, <img class="teximage" src="/files/tex/4aaf85facb6534fd470edd32dbdb4e28f6218190.png" alt="$ 3 $" /> appears twice, <img class="teximage" src="/files/tex/b1111b1328c0de162c7bb7bdbde0496f3f37f563.png" alt="$ 6 $" /> appears three times, and <img class="teximage" src="/files/tex/f9664c0e5fcffbbe88ed551c02b9275233d8f6b4.png" alt="$ 10 $" /> appears <img class="teximage" src="/files/tex/1f1498726bb4b7754ca36de46c0ccdd09136d115.png" alt="$ 4 $" /> times. There are infinite families of numbers known to appear <img class="teximage" src="/files/tex/b1111b1328c0de162c7bb7bdbde0496f3f37f563.png" alt="$ 6 $" /> times. The only number known to appear <img class="teximage" src="/files/tex/f7af773db5dd39209048c17f92c19e05dcf1f295.png" alt="$ 8 $" /> times is <img class="teximage" src="/files/tex/a4b629b3ff5d800526a8a5424044deedccf3e753.png" alt="$ 3003 $" />. It is not known whether any number appears more than <img class="teximage" src="/files/tex/f7af773db5dd39209048c17f92c19e05dcf1f295.png" alt="$ 8 $" /> times. The conjectured upper bound could be <img class="teximage" src="/files/tex/f7af773db5dd39209048c17f92c19e05dcf1f295.png" alt="$ 8 $" />; Singmaster thought it might be <img class="teximage" src="/files/tex/f9664c0e5fcffbbe88ed551c02b9275233d8f6b4.png" alt="$ 10 $" /> or <img class="teximage" src="/files/tex/b34db6486854df4b3bd55ab83aac3216def7b5af.png" alt="$ 12 $" />. See <a href="http://en.wikipedia.org/wiki/Singmaster's conjecture">Singmaster's conjecture</a>.</p> </tr></td> </table> </td> </tr> </table> Singmaster, David Pascal&#039;s triangle Number Theory Combinatorial Number Theory http://openproblemgarden.org/op/singmasters_conjecture#comment Fri, 15 Mar 2019 11:22:03 +0100 Zach Teitler 60034 at http://openproblemgarden.org