{"id":17050,"date":"2016-07-27T19:37:06","date_gmt":"2016-07-27T19:37:06","guid":{"rendered":"http:\/\/www.oracletutoring.ca\/blog\/?p=17050"},"modified":"2016-07-27T19:38:53","modified_gmt":"2016-07-27T19:38:53","slug":"number-theory-another-problem-from-dudleys-elementary-number-theory","status":"publish","type":"post","link":"https:\/\/www.oracletutoring.ca\/blog\/number-theory-another-problem-from-dudleys-elementary-number-theory\/","title":{"rendered":"Number theory: another problem from Dudley’s Elementary Number Theory"},"content":{"rendered":"

The tutor investigates a problem involving composite numbers.<\/h1>\n

For problem 4b, page 19, of his Elementary Number Theory<\/u> (second edition), Dudley invites the reader to prove there are infinite n such that both 6n-1 and 6n+1 are composite. (Composite means not prime.)<\/p>\n

The first such n I find is 20: 6(20)-1=119=7(17), while 6(20)+1=121=11(11). Now, adding a product of 7 and 11 (such as 77) to 20, we arrive at<\/p>\n

6(97)-1=6(20+77)-1=6(20)-1 + 6(77)=119+6(77), which must be divisible by 7, since 119 is. Therefore, it’s composite.<\/p>\n

Similarly,<\/p>\n

6(97)+1=6(20+77)+1=6(20)+1 + 6(77)=121 + 6(77), which must be divisible by 11, since 121 is.<\/p>\n

So, n=20+77k, where k is any integer, gives 6n-1, 6n+1 both composite.<\/p>\n

Source:<\/p>\n

Dudley, Underwood. Elementary Number Theory<\/u>, second ed. 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 investigates a problem involving composite numbers. For problem 4b, page 19, of his Elementary Number Theory (second edition), Dudley invites the reader to prove there are infinite n such that both 6n-1 and 6n+1 are composite. (Composite means …<\/p>\n

Number theory: another problem from Dudley’s Elementary Number Theory<\/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":[1753,1752,1751],"class_list":["post-17050","post","type-post","status-publish","format-standard","hentry","category-math","category-number-theory","tag-4b-page-19-elementary-number-theory","tag-composite-numbers","tag-underwood-dudley"],"_links":{"self":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/17050","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=17050"}],"version-history":[{"count":10,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/17050\/revisions"}],"predecessor-version":[{"id":17060,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/17050\/revisions\/17060"}],"wp:attachment":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/media?parent=17050"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/categories?post=17050"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/tags?post=17050"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}