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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}| =

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).

Solution Summary

Irrationalities and primes are manipulated.