Share
Explore BrainMass

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 &#966;n| &#966;m - 2

4) Prove that if n &#8800; m, then gcd (&#966;n, &#966;m) = 1

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

See the attached file.

Attachments

Solution Preview

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 &#8721; &#956;(d) &#966;(d) = &#928; (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 = &#966;n .
Prove that if n < m, then &#966;n| &#966;m - 2

4) Prove that if n &#8800; m, then gcd (&#966;n, &#966;m) = 1

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

Solution contains detailed step-by-step explanation.

$2.19