Describe EulerInformation here <\/p>\n
The totient function , also called Euler’s totient function, is defined as the number of positive integers that are relatively prime to (i.e., do not contain any factor in common with) , where 1 is counted as being relatively prime to all numbers. Since a number less than or equal to and relatively prime to a given number is called a totative, the totient function can be simply defined as the number of totatives of . For example, there are eight totatives of 24 (1, 5, 7, 11, 13, 17, 19, and 23), so . The totient function is implemented in Mathematica as EulerPhi[n]. <\/p>\n
is always even for . By convention, , although Mathematica defines EulerPhi[0] equal to 0 for consistency with its FactorInteger[0] command. The first few values of for , 2, … are 1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, … (Sloane’s A000010). The totient function is given by the Möbius transform of 1, 2, 3, 4, … (Sloane and Plouffe 1995, p. 22). is plotted above for small . <\/p>\n
For a prime , <\/p>\n
(1) <\/p>\n
since all numbers less than are relatively prime to . If is a power of a prime, then the numbers that have a common factor with are the multiples of : , , …, . There are of these multiples, so the number of factors relatively prime to is <\/p>\n
(2) <\/p>\n
Now take a general divisible by . Let be the number of positive integers not divisible by . As before, , , …, have common factors, so <\/p>\n
(3) <\/p>\n
Now let be some other prime dividing . The integers divisible by are , , …, . But these duplicate , , …, . So the number of terms that must be subtracted from to obtain is <\/p>\n
(4) <\/p>\n
and <\/p>\n
(5) (6) (7) <\/p>\n
By induction, the general case is then <\/p>\n
(8) <\/p>\n
An interesting identity relating to is given by <\/p>\n
(9) <\/p>\n
(A. Olofsson, pers. comm., Dec. 30, 2004). <\/p>\n
Another identity relates the divisors of to via <\/p>\n
(10) <\/p>\n","protected":false},"excerpt":{"rendered":"
Describe EulerInformation here The totient function , also called Euler’s totient function, is defined as the number of positive integers that are relatively prime to (i.e., do not contain any factor in common with) , where 1 is counted as being relatively prime to all numbers. Since a number less than or equal to and … \ub354 \uc77d\uae30<\/a><\/p>\n","protected":false},"author":7,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"ngg_post_thumbnail":0,"footnotes":""},"categories":[1],"tags":[],"class_list":["post-3366","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"distributor_meta":false,"distributor_terms":false,"distributor_media":false,"distributor_original_site_name":"\uae40\uc601\uc6b1","distributor_original_site_url":"https:\/\/mathematicians.korea.ac.kr\/ywkim","push-errors":false,"_links":{"self":[{"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/posts\/3366","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/comments?post=3366"}],"version-history":[{"count":1,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/posts\/3366\/revisions"}],"predecessor-version":[{"id":3367,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/posts\/3366\/revisions\/3367"}],"wp:attachment":[{"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/media?parent=3366"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/categories?post=3366"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/tags?post=3366"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}