Eulers Function s.t. k rel. prime to n Albert R Meyer March 9, - PowerPoint PPT Presentation

Euler φ function Mathematics for Computer Science MIT 6.042J/18.062J φ (n) ::= # k ∈ [0,n) 0,1 …,n-1 Euler’s Function s.t. k rel. prime to n Albert R Meyer March 9, 2012 lec 5F.1 Albert R Meyer March 9, 2012 lec 5F.2 Euler φ function Euler φ function gcd1{n} ::= φ (n) ::= # k ∈ [0,n) {k ∈ [0,n) | gcd(k,n) = 1} s.t. gcd(k,n) = 1 φ (n) = | gcd1{n} | so (some books write n* for gcd1{n}) Albert R Meyer March 9, 2012 lec 5F.3 Albert R Meyer March 9, 2012 lec 5F.4 1

Euler φ function Euler φ function gcd1{n} ::= gcd1{n} ::= {k ∈ [0,n) | gcd(k,n) = 1} {k ∈ [0,n) | gcd(k,n) = 1} gcd1{7} = {1,2,3,4,5,6} φ (7) = |{1,2,3,4,5,6}| gcd1{12} = gcd1{12} = {0,1,2,3,4,5,6,7,8,9,10,11} {0,1,2,3,4,5,6,7,8,9,10,11} Albert R Meyer March 9, 2012 lec 5F.5 Albert R Meyer March 9, 2012 lec 5F.6 Euler φ function Euler φ function gcd1{n} ::= gcd1{n} ::= {k ∈ [0,n) | gcd(k,n) = 1} {k ∈ [0,n) | gcd(k,n) = 1} φ (7) = 6 φ (7) = 6 gcd1{12} = φ (12) = {0,1,2,3,4,5,6,7,8,9,10,11} |{ 1, 5, 7, 11}| Albert R Meyer March 9, 2012 lec 5F.7 Albert R Meyer March 9, 2012 lec 5F.8 2

Euler φ function Calculating φ gcd1{n} ::= If p prime, everything in {k ∈ [0,n) | gcd(k,n) = 1} [1,p) is rel. prime to p, so φ (7) = 6 φ (p) = p – 1 φ (12) = 4 Albert R Meyer March 9, 2012 lec 5F.9 Albert R Meyer March 9, 2012 lec 5F.10 Calculating φ Calculating φ (p k ) φ (9)? 0,1,2,3,4,5,6,7,8 0,1,2,3,4,5,6,7,8 0 , 1 , ... , p , ... , 2p, ... ..,p k -p , ... , p k -1 0 , 1 , ... , p , ... , 2p, ... ..,p k -p , ... , p k -1 k rel. prime to 9 iff p divides every pth number k rel. prime to 3 p k /p of these numbers 3 divides every 3rd number are not rel. prime to p k so, φ (9) = 9-(9/3) = 6 Albert R Meyer March 9, 2012 lec 5F.11 Albert R Meyer March 9, 2012 lec 5F.12 3

Calculating φ (p k ) Calculating φ (p k ) so so φ (p k ) = p k - - p k /p φ (p k ) = p k - p k-1 Albert R Meyer March 9, 2012 lec 5F.13 Albert R Meyer March 9, 2012 lec 5F.14 Calculating φ (a ' Calculating φ (a ' b) ' b) Lemma :� φ (12) = φ (3 ' 4) � For a, b relatively prime, � = φ (3) ' φ (4) � φ (a ' b) = φ (a) ' φ (b) = (3 - 1) ' (2 2 - 2 2-1 ) pf: Pset 5. Another = 2 ' (4 - 2) = 4 way later by “counting.” Albert R Meyer March 9, 2012 lec 5F.15 Albert R Meyer March 9, 2012 lec 5F.16 4

Euler’s Theorem For k relatively prime to n, k φ (n) ≡ 1 (mod n) Albert R Meyer March 9, 2012 lec 5F.17 5

MIT OpenCourseWare http s ://ocw.mit.edu 6.042J / 18.062J Mathematics for Computer Science Spring 20 15 For information about citing these materials or our Terms of Use, visit: http s ://ocw.mit.edu/terms.