4. Let me N. Define the Euler totient function by o(m) = {k: 1 ≤ k <≤ m, gcd (k, m) = 1}], or the number of relatively prime positive integers less than or equal to m. Prove the following: ● (p) = p - 1 if p is prime ● (pk) = pk-pk-1 = pk (1) if p is prime and k ≥ 1 • If m, n are relatively prime, then o(mn) = o(m)o(n) 1 Deduce that if n = pi...p then (n)=n(-) (-) 1- ... Remark. The Euler totient function is an important function in number theory, c.f. Euler's Theorem.
4. Let me N. Define the Euler totient function by o(m) = {k: 1 ≤ k <≤ m, gcd (k, m) = 1}], or the number of relatively prime positive integers less than or equal to m. Prove the following: ● (p) = p - 1 if p is prime ● (pk) = pk-pk-1 = pk (1) if p is prime and k ≥ 1 • If m, n are relatively prime, then o(mn) = o(m)o(n) 1 Deduce that if n = pi...p then (n)=n(-) (-) 1- ... Remark. The Euler totient function is an important function in number theory, c.f. Euler's Theorem.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Combinatorics
![4. Let me N. Define the Euler totient function by (m) = {k: 1 ≤ k ≤
m, gcd (k, m) = 1}], or the number of relatively prime positive integers less than
or equal to m. Prove the following:
• (p) = p - 1 if p is prime
1
●
• (pk) = pk - pk-1 = pk (1-1) if p is prime and k ≥ 1
If m, n are relatively prime, then (mn) = o(m)o(n)
Deduce that if n = p¹...p then
ak
* (n) = n(1-21)... (1-21)
Remark. The Euler totient function is an important function in number
theory, c.f. Euler's Theorem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa14a0662-96d3-454f-9070-eb808829fccd%2F783e350d-447b-4b85-9ba0-1851a6d6c183%2Fj2378gh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4. Let me N. Define the Euler totient function by (m) = {k: 1 ≤ k ≤
m, gcd (k, m) = 1}], or the number of relatively prime positive integers less than
or equal to m. Prove the following:
• (p) = p - 1 if p is prime
1
●
• (pk) = pk - pk-1 = pk (1-1) if p is prime and k ≥ 1
If m, n are relatively prime, then (mn) = o(m)o(n)
Deduce that if n = p¹...p then
ak
* (n) = n(1-21)... (1-21)
Remark. The Euler totient function is an important function in number
theory, c.f. Euler's Theorem.
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