Math 5336 – Number Theory Day 12 Class activities for November 18,  2002

Dr. Cordero

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