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Mobius Function, Euler Function and Carmicheal's Conjecture

1) Prove that in is a positive integer. ( : is the Mobius function)
Hint: one of the four argument is divisible by 4.

2) If is a prime and . Show that ( : is the Euler function)

3) a. Prove that is an integer if n is a prime and that it is not an integer
b. Prove that is not an integer if n is divisible by the square of a prime.

4) Prove Carmichael's conjecture for each ( mod 4)
(Carmichael conjectured that for each integer n, there exists an integer m n such that (n) = (m)
( : is the Euler function)

Please see the attached file for the fully formatted problems.


Solution Summary

Mobius Functions, Euler Functions and Carmicheal's Conjecture are investigated. The solution is detailed and well presented.