# Proof of the Irrationality using Euler's phi-function

2. Apply the proof of the irrationality of sqrt(2) to a) sqrt(3) and b)

sqrt(4). If the proof breaks down, indicate precisely why.

3. Euler's phi-function is defined such that for n > 0, phi(n) = |{m <

n: gcd(m,n)=1}|. So, e.g., phi(4) = |{1,3}| = 2; phi(5) = |{1,2,3,4}| =

4.

a. Show that for prime p, phi(p) = p-1.

b. Show that for prime p and q, phi(p*q) = (p-1)*(q-1).

Â© BrainMass Inc. brainmass.com November 24, 2022, 11:32 am ad1c9bdddfhttps://brainmass.com/math/algebra/proof-irrationality-euler-sphi-function-5041

#### Solution Preview

Please see the attached file for the full solution.

Thanks for using BrainMass.

2. Apply the proof of the irrationality of sqrt(2) to a) sqrt(3) and b)

sqrt(4). If the proof breaks down, indicate precisely why.

Solution:

You haven't indicated which proof to use so I hope you now this one:

a)Suppose

is rational. This means that for some integers p and q

Square both sides

Multiply both sides by q2

By Fundamental Theorem of Arithmetic, each of ...

#### Solution Summary

Irrationalities and primes are manipulated. The proof of the irrationality using Euler's phi-functions.