Math 5336 Number Theory Dr. Cordero
Class activities for November 4, 2002
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 integersand , .
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.