Problem 39 - Discrete Mathematics with Applications

Question

Let \(n \in \mathbb{N}\) . A positive integer \(d\) is a proper factor of \(n\) if \(d\) is a factor of \(n\) and \(d < n .\) For example, the proper factors of 12 are \(1,2,3,4,\) and 6 . Let \(\sigma : N \rightarrow N\) defined by \(\sigma(n)=\) sum of the proper factors of \(n .\) (\sigma is the lowercase Greek letter, sigma.) Compute \(\sigma(n)\) for each value of \(n,\) where \(p\) and \(q\) are distinct primes, IA positive integer \(n\) such that \(\sigma(n)=n\) is a perfect number. $$p q$$

Step-by-Step Solution

Answer

Given a positive integer n = pq, where p and q are distinct prime numbers, compute the value of the function σ(n) = 1 + p + q. If σ(n) = n, then n is a perfect number.

Step 1: List the proper factors of n

For a positive integer n = pq, where p and q are distinct prime numbers, list all proper factors of n. We know that proper factors are divisors of n except for n itself. So, the proper factors of n are: 1 (trivial proper factor) p (since p is a prime number and a divisor of n) q (since q is another prime number and a divisor of n)

Step 2: Compute the value of σ(n)

Now that we have identified the proper factors of n, we can compute the value of σ(n), which refers to the sum of all proper factors. Thus, for an integer n = pq, the value of the function σ(n) can be calculated as follows: \[\sigma(n) = 1 + p + q\]

Step 3: Determine if σ(n) = n is a perfect number

A perfect number is defined as a positive integer in which the sum of its proper factors is equal to the integer itself. In other words, in our case, a positive integer n is a perfect number if: \[\sigma(n) = n\] Using the value of σ(n) computed in step 2, we will check if it is equal to the integer n: \[1 + p + q= pq\] If the given condition (1 + p + q = pq) is satisfied, then the positive integer n is a perfect number. Otherwise, it is not a perfect number. For example: Let n = 6, which is the product of distinct prime numbers p = 2 and q = 3. The proper factors of 6 are 1, 2, and 3. By computing σ(n), we get: \[\sigma(6) = 1 + 2 + 3 = 6\] Since σ(6) = 6 is satisfied, 6 is a perfect number.

Key Concepts

Proper Factors

A proper factor of a number is a divisor that is less than the number itself. This means if you take a number like 12, all divisors except the number itself are its proper factors. These include:

• 1
• 2
• 3
• 4
• 6

For the number 12, 12 is not included in its list of proper factors. Proper factors are crucial when determining perfect numbers, because a perfect number is one where the sum of all its proper factors equals the number itself. Understanding this helps us to evaluate whether a number can be considered perfect.

Distinct Primes

When we say numbers are distinct primes, it means each number is a prime and different from the other. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. Examples include 2, 3, 5, 7, etc. When calculating factors, distinct primes are important because they dictate the form of the number. For example, if a number is the product of distinct primes, such as 6 (which is 2 and 3), its structure is significantly different from numbers with repeated primes, like 9 (which is 3 times 3). These characteristics affect how we list proper factors and sum them in functions like the sigma function.

Sigma Function

The sigma function is a special mathematical function often represented by the Greek letter \( \sigma \). This function calculates the sum of the proper factors of a given integer. For instance, if \( n = pq \) (with \( p \) and \( q \) being distinct primes), the sigma function \( \sigma(n) \) would sum the proper factors of \( n \). Therefore, for \( n = 6 \), the proper factors being 1, 2, and 3, means \( \sigma(6) = 1 + 2 + 3 = 6 \). This simple computation checks whether a number is perfect if \( \sigma(n) = n \). Thus, every step with the sigma function moves us closer to determining if our number is a perfect number.

Divisors

Divisors of a number are those numbers that divide the original number without leaving a remainder. Each number has a set of divisors that includes itself and all smaller numbers that fit into it evenly. For example, consider the number 6. Its divisors include:

• 1
• 2
• 3
• 6

In the context of perfect numbers, we are especially interested in proper divisors, which are all divisors except the number itself. Knowing the proper divisors allows us to effectively use the sigma function and assess the number's perfect status. Understanding divisors is fundamental because they form the basis of other complex mathematical evaluations like factorization and prime checks.