2007-04-04 (Wednesday):
  1. Turn in graded version of hmwk today.  If you lost graded
     homework, tell me.  If you forgot to grade it, shame on you.
  2. Hand out new homework assignment -- reminder -- due on Monday!
  3. Questions?
  4. Define Li(x) and state RH:
         |pi(x) - Li(x)| <= sqrt(x)*log(x)
  5. Define group, abelian group, and ring.
  6. Define Z/nZ
  7. Define field.  Ex: Z/pZ
  8. Divisibility by 3 rule. 
  9. Prove cancellation (a*c = b*c ==> a = b)
 10. That any a with gcd(a,n)=1 is invertible modulo n.
     Proof: the map x |--> a*x is injective, since if 
     a*x = a*y then x==y by cancellation.  Therefore it
     is a bijection.  Done. 
 11. State but don't prove prop 2.1.13.
 12. Define phi(n).
 13. Prove Fermat's Little Theorem, that a^(phi(n)) = 1 (mod n)
     for any a with gcd(a,n) = 1.
     Proof: the map x |--> a*x is a bijection on the set of 
     [x] with gcd(x,n) = 1.  Thus prod x = prod a*x, so done.
  
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After class optional intro to SAGE:
  1. How to download and install: 
        - Linux
        - OS X
        - Windows
  2. Online notebook (access can be spotty)
  3. Start up and do a demo:
      - quo_rem
      - do the explicit calculation of Mersenne prime
      - gcd
      - prime_range
  4. General questions.