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3. Let be the Euler phi function. (a) Determine all integers n such that (n) is odd. (b) Determine all integers n such that o(n) = o(2n). (c) Prove that if r>1 and n > 0 are integers, then p(n") = n²-¹o(n).

3. Let be the Euler phi function. (a) Determine all integers n such that (n) is odd. (b) Determine all integers n such that o(n) = o(2n). (c) Prove that if r>1 and n > 0 are integers, then p(n") = n²-¹o(n).

Advanced Engineering Mathematics
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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3. Let be the Euler phi function. (a) Determine all integers n such that (n) is odd. (b) Determine all integers n such that o(n) = (2n). (c) Prove that if r> 1 and n> 0 are integers, then p(n") = n²-¹(n). (d) If m | n, show that o(m) (n). (e) Let (n) denote the sum of the p(d), as d varies over all positive divisors of n: Φ(n) - Σφ(α). d|n Calculate a few values of (n), and hence conjecture a formula for (n). By writing the n distinct fractions 1/n, 2/n, ..., n/n in lowest terms a/b, where ged (a, b) = 1 and bn, prove your conjecture.
Transcribed Image Text:3. Let be the Euler phi function. (a) Determine all integers n such that (n) is odd. (b) Determine all integers n such that o(n) = (2n). (c) Prove that if r> 1 and n> 0 are integers, then p(n") = n²-¹(n). (d) If m | n, show that o(m) (n). (e) Let (n) denote the sum of the p(d), as d varies over all positive divisors of n: Φ(n) - Σφ(α). d|n Calculate a few values of (n), and hence conjecture a formula for (n). By writing the n distinct fractions 1/n, 2/n, ..., n/n in lowest terms a/b, where ged (a, b) = 1 and bn, prove your conjecture.
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