Problem 61 - Discrete Mathematics with Applications

Question

Let \(t\) denote the tau function. Prove each. If \(m\) and \(n\) are relatively prime numbers, then \(\tau(m n)=t(m) \cdot t(n)\)

Step-by-Step Solution

Answer

In summary, by analyzing the prime factorizations of two relatively prime numbers \(m\) and \(n\), we have shown that the tau function of their product, \(t(mn)\), is equal to the product of their individual tau functions, \(t(m) \cdot t(n)\). This is demonstrated by calculating the numbers of divisors for \(m\), \(n\), and \(mn\) and observing that their factors are the same: \(t(mn) = t(m) \cdot t(n)\).

Step 1: Prime factorization of m and n

To analyze the divisors of \(m\) and \(n\), we first need to look at their prime factorizations. Since \(m\) and \(n\) are relatively prime, no prime factors are common between them. Let the prime factorization of \(m\) be: \(m = p_1^{a_1}p_2^{a_2}...p_k^{a_k}\) Let the prime factorization of \(n\) be: \(n = q_1^{b_1}q_2^{b_2}...q_l^{b_l}\) With \(\{p_i\}\) and \(\{q_j\}\) being prime numbers and their exponents being \(a_i\) and \(b_j\), respectively. From this, we have the product of \(m\) and \(n\) as: \(mn = p_1^{a_1}p_2^{a_2}...p_k^{a_k}q_1^{b_1}q_2^{b_2}...q_l^{b_l}\)

Step 2: Calculate the divisors of m and n

Now we will calculate the number of divisors of \(m\) and \(n\) using their respective prime factorizations. The number of divisors of a number is calculated as the product of one more than each of its prime factor exponents. For \(m\), the number of divisors is: \(t(m) = (a_1 + 1)(a_2 + 1)...(a_k + 1)\) For \(n\), the number of divisors is: \(t(n) = (b_1 + 1)(b_2 + 1)...(b_l + 1)\)

Step 3: Calculate the divisors of the product mn

We'll now calculate the divisors of the product \(mn\). For \(mn\), the number of divisors is: \(t(mn) = (a_1 + 1)(a_2 + 1)...(a_k + 1)(b_1 + 1)(b_2 + 1)...(b_l + 1)\)

Step 4: Prove the relationship between divisors of m, n, and mn

We need to show that \(t(mn) = t(m) \cdot t(n)\). From Step 2, we have the number of divisors for \(m\) and \(n\): \(t(m) = (a_1 + 1)(a_2 + 1)...(a_k + 1)\) \(t(n) = (b_1 + 1)(b_2 + 1)...(b_l + 1)\) From Step 3, we have the number of divisors for \(mn\): \(t(mn) = (a_1 + 1)(a_2 + 1)...(a_k + 1)(b_1 + 1)(b_2 + 1)...(b_l + 1)\) Now we can see that both \(t(mn)\) and the product of \(t(m)\) and \(t(n)\) have the same set of factors: \(t(mn) = t(m) \cdot t(n) = [(a_1 + 1)(a_2 + 1)...(a_k + 1)][(b_1 + 1)(b_2 + 1)...(b_l + 1)]\) Hence, we have proven that for any two relatively prime numbers \(m\) and \(n\), their respective tau functions and the tau function of their product follow the relationship: \(t(mn) = t(m) \cdot t(n)\).

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a composite number into a product of its prime factors. This step is crucial because prime numbers are the building blocks of all natural numbers, and knowing a number's prime factors can help us understand its properties, such as how many divisors it has.

For example, let's consider the number 28. The prime factorization of 28 is:
\[28 = 2^2 \times 7^1\]
Here, we see that 28 is made up of the prime numbers 2 and 7. The exponent 2 on the prime factor 2 denotes that 2 is used twice in the multiplication. Similarly for larger numbers, prime factorization would reveal all the prime numbers and their powers that multiply together to create the number.

Relatively Prime Numbers

Relatively prime numbers, also known as coprime numbers, are pairs of numbers that have no common prime factors other than 1. In other words, their greatest common divisor (GCD) is 1. This concept is fundamental in the study of number theory because it influences many other properties and operations involving numbers.

For instance, consider the numbers 8 and 15. The prime factorization of 8 is \(2^3\), and for 15 it is \(3^1 \times 5^1\). Since there are no common prime factors between them, we say 8 and 15 are relatively prime. This relationship plays a key role, especially in the behavior of certain functions and theorems in mathematics, such as Euler's Totient function and the topic of today's article, the tau function.

Number of Divisors

The number of divisors of a natural number can be determined once we know its prime factorization. The tau function, denoted by \(t(n)\), represents the number of divisors of the number \(n\). If \(n\) has the prime factorization \(p_1^{a_1}p_2^{a_2}...p_k^{a_k}\), then the number of divisors, or \(t(n)\), is found by multiplying together one more than each of the exponents of the prime factors.

For example, for a number with prime factorization \(2^3 \times 3^2\), the number of divisors would be calculated as:
\[t(n) = (3 + 1)(2 + 1) = 4 \times 3 = 12\]
Thus, there are 12 divisors of the number. This calculation comes in handy when exploring the tau function, particularly in the case of two relatively prime numbers and their product, as seen in the earlier provided exercise.