Math 5336 Number Theory                                                                Dr. Cordero

Class activities for November 18,  2002

Topic: Number Theoretic Functions

1.      Definitions:

• Given a positive integer n, let  denote the number of positive divisors on n and  denote the sum of these divisors.
• For any arbitrary real number x, we denote by [x] the largest integer less than or equal to x; that is, [x] is the unique integer satisfying x-1 <[x] < x.

2. Compute  for each given n:

§         n=75

§         n=99

§         n=243

§         n=1024

3. What do you think  is, given that p and q are distinct primes?

What do you think  is, given that p and q are distinct primes?

What is ?

What is ?

4. Study Theorem 6.1 Formula to obtain all the positive divisors of an integer n

Theorem  6.2  Formulas for  and

Theorem 6.3   If gcd(a, b)=1, then  and

3. Practice:

§         Find  by means of the formula given in Theorem 6.2:

1.      n=75

2.      n=900

3.      n=6961

4.      n=17,640

§         Page 109 # 5a

§         Page 109 # 7a: Prove that  is an odd integer if and only if n is a perfect square.

4. An application to the calendar: Read § 6.4. Then do Problems # 1, 2, 3, 5, 6.