Math 5336 – Number Theory Day 10 Class activities for November 4, 2002

Dr. Cordero

Topic: Euler’s Phi Function, II

  1. Come up with a conjecture of the form
    “There is a positive integer k such that (mod m) if and only if (some condition on a and m).
  2.  Euler Phi Function: For  , the value of the Euler Phi Function is defined to be   and gcd.
  3.  Compute  .
  4. Prove the following theorems:
    • Theorem:  If p is prime, .
    • Theorem: If p is prime  .
    • Theorem: If p is prime and _  is an integer, then  _.
    • Theorem: If p and q are primes with _, then  _.
  5. Theorem 7.2: For any positive integers _ and _ , _ .
    Theorem 7.3: If the integer n>1 has the prime factorization _ , then _.
    (At-Home Practice: Study the proof of these theorems, pp. 131-132)
  6. Prove the following: Theorem: For n>2, _ is an even integer.
  7. Practice: Page 133 # 2, 4
  8. At-Home Practice: Page 133 # 1, 5, 6.