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.