{"id":225,"date":"2019-02-01T15:21:43","date_gmt":"2019-02-01T21:21:43","guid":{"rendered":"http:\/\/blogs.acu.edu\/mu_sigma\/?p=225"},"modified":"2019-02-08T14:58:36","modified_gmt":"2019-02-08T20:58:36","slug":"problem-11-when-are-there-integer-solutions-february-1-2019","status":"publish","type":"post","link":"https:\/\/blogs.acu.edu\/mu_sigma\/2019\/02\/01\/problem-11-when-are-there-integer-solutions-february-1-2019\/","title":{"rendered":"Problem 11 &#8211; When Are There Integer Solutions &#8211; Feb 1, 2019"},"content":{"rendered":"<h4><b>Problem 11 \u2013 February 1st, 2019<\/b><\/h4>\n<p><span style=\"font-size: xx-large\"><b>When Are There Integer Solutions?<\/b><\/span><\/p>\n<h4>Describe all positive integers <i>a<\/i> for which the equation <i>x<\/i><sup>2<\/sup> &#8211; <i>y<\/i><sup>2<\/sup> = <i>a<\/i> has integer solutions.<\/h4>\n<p>&nbsp;<\/p>\n<p>Submit your answers to mathpotw@acu.edu. \u00a0Details for submissions can be found\u00a0<a href=\"http:\/\/blogs.acu.edu\/mu_sigma\/2018\/08\/28\/problem-of-the-week-competition\/\">here<\/a>.<\/p>\n<p><em>Correct solutions were submitted by Wyatt Witemeyer.<\/em><\/p>\n<p><span style=\"font-size: large\">Solution to <b>When Are There Integer Solutions?<\/b><\/span><\/p>\n<p>The expression <i>x<\/i><sup>2<\/sup> &#8211; <i>y<\/i><sup>2<\/sup> factors into the product (<i>x<\/i> &#8211; <i>y<\/i>)(<i>x<\/i> + <i>y<\/i>). Now <i>x<\/i> &#8211; <i>y<\/i> and <i>x<\/i> + <i>y<\/i> have the same parity, that is, they are either both even or both odd. If they are both even, then their product is a multiple of 4. If they are both odd, then their product is odd.<\/p>\n<p>Thus the positive integer <i>a<\/i> must either be odd or be a multiple of 4.<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Problem 11 \u2013 February 1st, 2019 When Are There Integer Solutions? Describe all positive integers a for which the equation x2 &#8211; y2 = a has integer solutions. &nbsp; Submit your answers to mathpotw@acu.edu. \u00a0Details for submissions can be found\u00a0here. Correct solutions were submitted by Wyatt Witemeyer. Solution to When Are There Integer Solutions? The [&hellip;]<\/p>\n","protected":false},"author":130,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_et_pb_use_builder":"","_et_pb_old_content":"","_et_gb_content_width":"","footnotes":""},"categories":[181534],"tags":[],"class_list":["post-225","post","type-post","status-publish","format-standard","hentry","category-problem-of-the-week"],"_links":{"self":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/posts\/225","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/users\/130"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/comments?post=225"}],"version-history":[{"count":4,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/posts\/225\/revisions"}],"predecessor-version":[{"id":233,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/posts\/225\/revisions\/233"}],"wp:attachment":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/media?parent=225"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/categories?post=225"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/tags?post=225"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}