# The Mobius Inversion Formula

Theory of Numbers

Mobius Theorem

The Mobius Inversion Formula

1) Prove that ∑ μ(d) φ(d) = Π (2 - p)

d/n p/n

Primitive roots modulo p

2) Find all primitive roots modulo 5, modulo 9, modulo 11, modulo 13 and modulo 15.

Prime Numbers

3) The Fermat numbers are numbers of the form 2^(2^n) + 1 = φn .

Prove that if n < m, then φn| φm - 2

4) Prove that if n ≠ m, then gcd (φn, φm) = 1

5) Use the above exercise to give a proof that there exist infinitely many primes.

See the attached file.

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Theory of Numbers

Mobius Theorem

The ...

#### Solution Summary

This solution is comprised of a detailed explanation of the Mobius Inversion Formula,

Primitive roots modulo p and Fermat numbers.

It contains step-by-step explanation for the following problem:

The Mobius Inversion Formula

1) Prove that ∑ μ(d) φ(d) = Π (2 - p)

d/n p/n

Primitive roots modulo p

2) Find all primitive roots modulo 5, modulo 9, modulo 11, modulo 13 and modulo 15.

Prime Numbers

3) The Fermat numbers are numbers of the form 2^(2^n) + 1 = φn .

Prove that if n < m, then φn| φm - 2

4) Prove that if n ≠ m, then gcd (φn, φm) = 1

5) Use the above exercise to give a proof that there exist infinitely many primes.

Solution contains detailed step-by-step explanation.