{"id":16334,"date":"2016-06-27T23:57:52","date_gmt":"2016-06-27T23:57:52","guid":{"rendered":"http:\/\/www.oracletutoring.ca\/blog\/?p=16334"},"modified":"2016-06-27T23:59:08","modified_gmt":"2016-06-27T23:59:08","slug":"math-number-theory-a-formula-for-generating-pythagorean-triples","status":"publish","type":"post","link":"https:\/\/www.oracletutoring.ca\/blog\/math-number-theory-a-formula-for-generating-pythagorean-triples\/","title":{"rendered":"Math: number theory: a formula for generating Pythagorean triples"},"content":{"rendered":"

The tutor continues his discussion about Pythagorean triples.<\/h1>\n

Back in my January 7, 2016 post<\/a> I brought up Pythagorean triples, which are all-integer solutions to<\/p>\n

x² + y² = z²<\/p>\n

The equation above is based on the familiar<\/p>\n

a² + b² = c²<\/p>\n

An interesting fact is that Pythagorean triples can be generated from the following formulas, with n odd:<\/p>\n\n\n\n\n
x<\/td>\nn<\/td>\n<\/tr>\n
y<\/td>\n(n²-1)\/2<\/td>\n<\/tr>\n
z<\/td>\n(n²+1)\/2<\/td>\n<\/tr>\n<\/table>\n

<\/p>\n

Take, for instance, n as 3. Then we have<\/p>\n

x=3<\/p>\n

y=(3²-1)\/2=(9-1)\/2=8\/2=4 3(2)\/2 + 2\/2=4<\/p>\n

z=(3²+1)\/2=(9+1)\/2=10\/2=5 <\/p>\n

which is the familiar 3,4,5 triple.<\/p>\n

With n=11, we have<\/p>\n

x=11<\/p>\n

y=(11²-1)\/2=(120)\/2=60<\/p>\n

z=(11²+1)\/2=(122)\/2=61<\/p>\n

Checking, we start with<\/p>\n

11²+60²=61²<\/p>\n

Squaring, we get<\/p>\n

121+3600=3721✓<\/p>\n

I’ll be giving more coverage to Pythagorean triples in coming posts:)<\/p>\n

Source:<\/p>\n

Dudley, Underwood. Elementary Number Theory<\/u>. New York:
  W 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 continues his discussion about Pythagorean triples. Back in my January 7, 2016 post I brought up Pythagorean triples, which are all-integer solutions to x² + y² = z² The equation above is based on the familiar a² + …<\/p>\n

Math: number theory: a formula for generating Pythagorean triples<\/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":[1678,1328],"class_list":["post-16334","post","type-post","status-publish","format-standard","hentry","category-math","category-number-theory","tag-formula-for-generating-pythagorean-triples","tag-pythagorean-triples"],"_links":{"self":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/16334","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=16334"}],"version-history":[{"count":34,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/16334\/revisions"}],"predecessor-version":[{"id":16368,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/16334\/revisions\/16368"}],"wp:attachment":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/media?parent=16334"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/categories?post=16334"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/tags?post=16334"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}