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Mobius Function, Euler Function and Carmicheal's Conjecture

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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.

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Mobius functions, euler functions and Carmicheal's conjecture are investigated. The solution is detailed and well presented.

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