{"id":16369,"date":"2016-06-28T19:33:39","date_gmt":"2016-06-28T19:33:39","guid":{"rendered":"http:\/\/www.oracletutoring.ca\/blog\/?p=16369"},"modified":"2018-02-13T14:52:43","modified_gmt":"2018-02-13T14:52:43","slug":"math-pythagorean-triples-proof-of-yesterdays-generating-formulas","status":"publish","type":"post","link":"https:\/\/www.oracletutoring.ca\/blog\/math-pythagorean-triples-proof-of-yesterdays-generating-formulas\/","title":{"rendered":"Math: Pythagorean triples: proof of yesterday’s generating formulas"},"content":{"rendered":"

The tutor shows that yesterday’s formulas to generate Pythagorean triples are valid.<\/h1>\n

In yesterday’s post<\/a> I showed a way to generate Pythagorean triples x, y, z from an odd number n:<\/p>\n\n\n\n\n
x<\/td>\nn<\/td>\n<\/tr>\n
y<\/td>\n(n\u00b2-1)\/2<\/td>\n<\/tr>\n
z<\/td>\n(n\u00b2+1)\/2<\/td>\n<\/tr>\n<\/table>\n

Let’s make sure that, generated as above, x\u00b2+y\u00b2=z\u00b2<\/p>\n

n^2 +((n^2 -1)\/2)^2 = n^2 + (n^4-2n^2+1)\/4<\/p>\n

which leads to, after getting a common denominator,<\/p>\n

4n^2\/4 + (n^4-2n^2+1)\/4 = (n^4+2n^2+1)\/4<\/p>\n

Note also that<\/p>\n

((n^2+1)\/2)^2 = (n^4+2n^2+1)\/4<\/p>\n

Therefore<\/p>\n

n^2 +((n^2 -1)\/2)^2 = ((n^2+1)\/2)^2<\/p>\n

so the generating formula for pythagorean triples is proven.<\/p>\n

Source:<\/p>\n

Dudley, Underwood. Elementary Number Theory<\/u>. New York:
\u00a0\u00a0W H Freeman and Company, 1978.<\/p>\n

Jack of Oracle Tutoring by Jack and Diane,<\/a> Campbell River, BC.<\/p>\n","protected":false},"excerpt":{"rendered":"

The tutor shows that yesterday’s formulas to generate Pythagorean triples are valid. In yesterday’s post I showed a way to generate Pythagorean triples x, y, z from an odd number n: x n y (n\u00b2-1)\/2 z (n\u00b2+1)\/2 Let’s make sure …<\/p>\n

Math: Pythagorean triples: proof of yesterday’s generating formulas<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3,1677],"tags":[1678,1679,1328],"class_list":["post-16369","post","type-post","status-publish","format-standard","hentry","category-math","category-number-theory","tag-formula-for-generating-pythagorean-triples","tag-proof-of-formula-for-generating-pythagorean-triples","tag-pythagorean-triples"],"_links":{"self":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/16369","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/comments?post=16369"}],"version-history":[{"count":32,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/16369\/revisions"}],"predecessor-version":[{"id":29600,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/16369\/revisions\/29600"}],"wp:attachment":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/media?parent=16369"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/categories?post=16369"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/tags?post=16369"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}