{"id":1218,"date":"2019-08-20T08:51:11","date_gmt":"2019-08-20T12:51:11","guid":{"rendered":"http:\/\/josmfs.net\/?p=1218"},"modified":"2020-11-12T11:08:42","modified_gmt":"2020-11-12T16:08:42","slug":"consecutive-product-square","status":"publish","type":"post","link":"https:\/\/josmfs.net\/wordpress\/2019\/08\/20\/consecutive-product-square\/","title":{"rendered":"Consecutive Product Square"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft  wp-image-1216\" src=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2019\/08\/Consecutive-Product-Square-Fig2.jpg\" alt=\"\" width=\"231\" height=\"27\" \/>This problem from Colin Hughes at <em>Maths Challenge<\/em> is a most surprising result that takes a bit of tinkering to solve.<\/p>\n<p>\u201c<strong>Problem<\/strong><br \/>We can see that 3 x 4 x 5 x 6 = 360 = 19\u00b2 \u2013 1. Prove that the product of four consecutive integers is always one less than a perfect square.\u201d<\/p>\n<p>The result is so mysterious at first that you begin to understand why the ancient Pythagoreans had a mystical relationship with mathematics.<\/p>\n<p>See the <a href=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2019\/08\/Consecutive-Product-Square-190722.pdf\">Consecutive Product Square<\/a>.<\/p>\n<p>(<strong>Update 11\/12\/2020<\/strong>) <strong>Generalization and Visual Proof <br \/><\/strong><!--more--><\/p>\n<p>From <a href=\"https:\/\/twitter.com\/fermatslibrary\/status\/1323996949410992136\">Fermat&#8217;s Library<\/a> on 4 November 2020 we have a visual proof of a generalization of the consecutive integer problem:<\/p>\n<p>\u201cA visual proof that the product of 4 positive integers in arithmetic progression is the difference of two squares<\/p>\n<p style=\"text-align: center;\">n(n + d)(n + 2d)(n + 3d) = (n\u00b2 + 3nd + d\u00b2)\u00b2 \u2013 (d\u00b2)\u00b2 \u201d<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-large wp-image-1843 aligncenter\" src=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2020\/11\/Consecutive-Product-Square-Fig-10a-1024x357.jpg\" alt=\"\" width=\"625\" height=\"218\" srcset=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2020\/11\/Consecutive-Product-Square-Fig-10a-1024x357.jpg 1024w, https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2020\/11\/Consecutive-Product-Square-Fig-10a-300x105.jpg 300w, https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2020\/11\/Consecutive-Product-Square-Fig-10a-768x268.jpg 768w, https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2020\/11\/Consecutive-Product-Square-Fig-10a-624x217.jpg 624w, https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2020\/11\/Consecutive-Product-Square-Fig-10a.jpg 1200w\" sizes=\"auto, (max-width: 625px) 100vw, 625px\" \/><\/p>\n<p><a href=\"https:\/\/josmfs.net\/wordpress\/consecutive-product-square-gif\/\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1839 aligncenter\" src=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2020\/11\/Consecutive-Product-Square-Fig-GIF.jpg\" alt=\"\" width=\"432\" height=\"247\" srcset=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2020\/11\/Consecutive-Product-Square-Fig-GIF.jpg 432w, https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2020\/11\/Consecutive-Product-Square-Fig-GIF-300x172.jpg 300w\" sizes=\"auto, (max-width: 432px) 100vw, 432px\" \/><\/a><\/p>\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>This problem from Colin Hughes at Maths Challenge is a most surprising result that takes a bit of tinkering to solve. \u201cProblemWe can see that 3 x 4 x 5 x 6 = 360 = 19\u00b2 \u2013 1. Prove that the product of four consecutive integers is always one less than a perfect square.\u201d The [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[133,151],"class_list":["post-1218","post","type-post","status-publish","format-standard","hentry","category-puzzles-and-problems","tag-algebra","tag-colin-hughes"],"_links":{"self":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/1218","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/comments?post=1218"}],"version-history":[{"count":5,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/1218\/revisions"}],"predecessor-version":[{"id":1844,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/1218\/revisions\/1844"}],"wp:attachment":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/media?parent=1218"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/categories?post=1218"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/tags?post=1218"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}