This assignment is due next Thursday (10/22) but (as usual) start it over the weekend so we can discuss it some on Tuesday.

1. Show that 2, 3, a=1+sqrt(-5) and b=1-sqrt(-5) are all irreducible in R = Z(sqrt(-5)), the set of all numbers of the form a+b*sqrt(-5) for integers a and b. Since 6 = 2*3 = a*b and a is not a unit times 2 or 3, the fundamental theorem of arithmetic fails in R.

   Hint. Use the fact that for complex numbers, |zw|² = |z|²|w|².

2. Show that unique factorization seriously fails in the ring Z(sqrt(-17)), the set of all numbers of the form a+b*sqrt(-17) for integers a and b.

   (By "seriously" I mean that even the number of irreducible factors can change from one factorization to another.)

3. Find (or look up and rewrite in your own words) a proof that Euler's phi function is multiplicative.

4. Show that for fixed n the function f(x) = gcd(x,n) is multiplicative.

5. Find all the integers n for which phi(n)=24.

6. When is phi(n) a power of 2?

7. Let g(n) be the number of integers k between 1 and n such that gcd(k,n) = gcd(k+1,n) = 1. Figure out as much as you can about g: calculations, conjectures, formulas, proofs.