# 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).

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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.