{"id":54,"date":"2004-12-12T14:26:03","date_gmt":"2004-12-12T14:26:03","guid":{"rendered":"http:\/\/www.sixthform.info\/maths\/?p=54"},"modified":"2004-12-12T14:26:03","modified_gmt":"2004-12-12T14:26:03","slug":"2-unrelated-problems","status":"publish","type":"post","link":"https:\/\/www.sixthform.info\/maths\/?p=54","title":{"rendered":"2 unrelated problems"},"content":{"rendered":"<ol>\n<li>Using the <a href=\"http:\/\/en.wikipedia.org\/wiki\/Gelfond-Schneider_theorem\" target=\"_blank\">Gelfond-Schneider theorem<\/a> show that if <img src='\/maths\/latexrender\/pictures\/e1671797c52e15f763380b45e841ec32.gif' title='e' alt='e' align=absmiddle> is <a href=\"http:\/\/en.wikipedia.org\/wiki\/Transcendental_number\" target=\"_blank\">transcendental<\/a> then <img src='\/maths\/latexrender\/pictures\/4f08e3dba63dc6d40b22952c7a9dac6d.gif' title='\\pi' alt='\\pi' align=absmiddle> is transcendental<\/li>\n<li>If <img src='\/maths\/latexrender\/pictures\/b345e1dc09f20fdefdea469f09167892.gif' title='a,b' alt='a,b' align=absmiddle> are positive integers such that <img src='\/maths\/latexrender\/pictures\/46fa7900cc397f3a4b3fa2e72d6885e7.gif' title='a&gt;b' alt='a&gt;b' align=absmiddle> and <img src='\/maths\/latexrender\/pictures\/65c884f742c8591808a121a828bc09f8.gif' title='a+b' alt='a+b' align=absmiddle> is even, show that <img src='\/maths\/latexrender\/pictures\/b98fd3ba951018248eda569e18f6287d.gif' title='a^2-a-b^2' alt='a^2-a-b^2' align=absmiddle> is not a square<\/li>\n<\/ol>\n<p><u>Notes<\/u><\/p>\n<ol>\n<li>This theorem was being discussed at a seminar many years ago as was the transcendence of <img src='\/maths\/latexrender\/pictures\/e1671797c52e15f763380b45e841ec32.gif' title='e' alt='e' align=absmiddle>. When we were told that it was an obvious consequence that <img src='\/maths\/latexrender\/pictures\/4f08e3dba63dc6d40b22952c7a9dac6d.gif' title='\\pi' alt='\\pi' align=absmiddle> is transcendental we thought that the speaker was joking. When he showed how it followed, there was a moment&#8217;s silence and then applause &#8211; one of those moments when one realises that mathematics is such a wonderful subject.<br \/>\nYou can find an elementary proof (not using Gelfond-Schneider) of the transcendence of <img src='\/maths\/latexrender\/pictures\/e1671797c52e15f763380b45e841ec32.gif' title='e' alt='e' align=absmiddle> and <img src='\/maths\/latexrender\/pictures\/4f08e3dba63dc6d40b22952c7a9dac6d.gif' title='\\pi' alt='\\pi' align=absmiddle> <a href=\"http:\/\/www.mayer.dial.pipex.com\/samples\/pi\/tex4ht\/pisample.html\" target=\"_blank\">here<\/a><br \/>\nIt&#8217;s interesting to note that <img src='\/maths\/latexrender\/pictures\/c883fb68a3a1f0754f59998d33409b3d.gif' title='e^{\\pi}' alt='e^{\\pi}' align=absmiddle> and <img src='\/maths\/latexrender\/pictures\/77f2e8bd51d6c5d33bd9b2dec137810e.gif' title='\\pi^e' alt='\\pi^e' align=absmiddle> are very close. In fact <img src='\/maths\/latexrender\/pictures\/4097581cc1cab172a65e514bcaf75ed6.gif' title='e^\\pi-\\pi^e \\approx 0.68' alt='e^\\pi-\\pi^e \\approx 0.68' align=absmiddle>. See <a href=\"http:\/\/www.sixthform.info\/maths\/b2commentspopup.php?p=48&#038;c=1\" target=\"_blank\">here<\/a><\/li>\n<li>I came across this problem when looking through some old papers. I have no idea where it comes from (an old Mathematical Olympiad problem maybe??) or what the solution was.<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Using the Gelfond-Schneider theorem show that if is transcendental then is transcendental If are positive integers such that and is even, show that is not a square Notes This theorem was being discussed at a seminar many years ago as was the transcendence of . When we were told that it was an obvious consequence [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-54","post","type-post","status-publish","format-standard","hentry","category-articles"],"_links":{"self":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/54","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=54"}],"version-history":[{"count":0,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/54\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=54"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=54"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=54"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}