https://en.formulasearchengine.com/index.php?title=St%C3%B8rmer_number&feed=atom&action=history Størmer number - Revision history 2020-09-23T09:06:53Z Revision history for this page on the wiki MediaWiki 1.36.0-alpha https://en.formulasearchengine.com/index.php?title=St%C3%B8rmer_number&diff=21778&oldid=prev en>Maxal at 16:50, 11 October 2013 2013-10-11T16:50:42Z <p></p> <p><b>New page</b></p><div>In mathematics, a '''Størmer number''' or '''arc-cotangent irreducible number''', named after [[Carl Størmer]], is a positive integer ''n'' for which the greatest prime factor of ''n''&lt;sup&gt;2&lt;/sup&gt;&amp;nbsp;+&amp;nbsp;1 meets or exceeds 2''n''. <br /> <br /> The first few Størmer numbers are:<br /> : [[1 (number)|1]], [[2 (number)|2]], [[4 (number)|4]], [[5 (number)|5]], [[6 (number)|6]], [[9 (number)|9]], [[10 (number)|10]], [[11 (number)|11]], [[12 (number)|12]], [[14 (number)|14]], [[15 (number)|15]], [[16 (number)|16]], [[19 (number)|19]], [[20 (number)|20]], ... {{OEIS|id=A005528}}. <br /> Todd proved that this sequence is [[Infinite set|infinite]] (but not [[Cofiniteness|cofinite]]).<br /> <br /> The Størmer numbers arise in connection with the problem of representing the [[Gregory number]]s ([[arctangent]]s of [[rational number]]s) &lt;math&gt;G_{a/b}=\arctan\frac{b}{a}&lt;/math&gt; as sums of Gregory numbers for integers (arctangents of [[unit fraction]]s). The Gregory number &lt;math&gt;G_{a/b}&lt;/math&gt; may be decomposed by repeatedly multiplying the [[Gaussian integer]] &lt;math&gt;a+bi&lt;/math&gt; by numbers of the form &lt;math&gt;n\pm i&lt;/math&gt;, in order to cancel prime factors ''p'' from the imaginary part; here &lt;math&gt;n&lt;/math&gt; is chosen to be a Størmer number such that &lt;math&gt;n^2+1&lt;/math&gt; is divisible by &lt;math&gt;p&lt;/math&gt;.&lt;ref&gt;Conway &amp; Guy (1996): 245, &amp;para; 3&lt;/ref&gt;<br /> <br /> ==Notes==<br /> {{reflist}}<br /> <br /> ==References==<br /> * [[John H. Conway]] &amp; [[R. K. Guy]], ''The Book of Numbers''. New York: Copernicus Press (1996): 245–248. <br /> * [[J. Todd]], &quot;A problem on arc tangent relations&quot;, ''Amer. Math. Monthly'', '''56''' (1949): 517–528.<br /> <br /> {{Classes of natural numbers}}<br /> {{DEFAULTSORT:Stormer Number}}<br /> [[Category:Integer sequences]]</div> en>Maxal