Problem 45 - Discrete Mathematics with Applications

Question

Prime numbers of the form \(f(n)=2^{n}-1,\) where \(n\) is a positive integer, are called Mersenne primes, after the Franciscan monk Marin Mersenne \((1588-1648) .\) For example, \(f(2)=3, f(3)=7,\) and \(f(5)=31\) are Mersenne primes. Give a counterexample to disprove the claim that if \(n\) is a prime, then \(2^{n}-1\) is a prime.

Step-by-Step Solution

Answer

Using the prime number n = 11, we can calculate \(f(11) = 2^{11} - 1 = 2047\). However, 2047 is not a prime number, as it is divisible by 23 and 89 (23 * 89 = 2047). This counterexample disproves the claim that if n is prime, then \(2^n - 1\) must also be prime.

Step 1: Identify a prime number n

Start with a prime number and evaluate f(n). Since the claim states that for any prime n, f(n) must also be prime, finding just one counterexample will disprove the claim. Let's start with a prime number n = 11.

Step 2: Calculate f(n) for the chosen prime number

Now, we need to determine the value of f(11), where f(n) = 2^n - 1. In this case, n = 11, so we calculate \(f(11) = 2^{11} - 1\).

Step 3: Evaluate the expression

Using the given function, we have: \(f(11) = 2^{11} - 1 = 2048 - 1 = 2047\)

Step 4: Determine if the result is prime or composite

The next step is to determine if 2047 is a prime number or not. A prime number is divisible only by 1 and itself. To check this, we can look for any factors of 2047. In this case, 2047 is divisible by 23 and 89 (23*89 = 2047). As 2047 has factors other than 1 and itself, it is a composite number, not a prime number.

Step 5: Conclusion

Since we've found a prime number n (n = 11) for which f(n) is not prime (f(11) = 2047, which is divisible by 23 and 89), we have successfully disproven the claim that if n is prime, then 2^n - 1 must also be prime.

Key Concepts

Prime Numbers

Prime numbers are the building blocks of mathematics, especially in the domain of number theory. A prime number is defined as a positive integer greater than 1 that has no divisors other than 1 and itself. This means they cannot be formed by multiplying two smaller numbers together.

Examples of prime numbers include 2, 3, 5, 7, 11, and so forth. The number 2 is particularly notable as it is the only even prime number; all other primes are odd, since any even number greater than 2 can be divided by 2.

Understanding prime numbers is crucial because they play a central role in various mathematical concepts, including the Fundamental Theorem of Arithmetic, which states that every integer greater than 1 either is a prime itself or can be factored into primes in a way that is unique up to the order of the factors.

Prime numbers are not just a theoretical concept but also have practical applications, such as in cryptography where they are used to encrypt information, underscoring their importance beyond pure mathematics.

Discrete Mathematics

Discrete mathematics is an area of mathematics that is concerned with discrete, as opposed to continuous, quantities. It deals with objects that can only take distinct, separated values. This field of study encompasses a wide range of topics including logic, set theory, graph theory, and combinatorics.

Prime numbers, as an example, are studied within discrete mathematics because they are individual, countable entities. Discrete mathematics provides the tools and principles for analyzing and understanding properties of integers, such as proving whether or not a number is prime.

One vital aspect of discrete mathematics is its emphasis on proof and logical reasoning. Proving conjectures, whether through direct proofs, proof by contradiction, or constructing counterexamples, is at the heart of this discipline. This process of rigorous verification is critical for establishing the validity of mathematical statements.

The relevance of discrete mathematics extends into computer science, where it has applications in algorithm design, analysis of algorithms, and complexity theory. It helps us to design algorithms that are both efficient and reliable, as in encryption algorithms that rely on prime numbers.

Counterexample

A counterexample is a specific example or case that demonstrates a statement to be false. In mathematics, when a claim is made, it often holds generally. However, if you can find even one situation where the claim does not hold, then the claim cannot be universally true.

For instance, the claim that 'if n is a prime number, then 2^n - 1 is also a prime' can be disproven by providing a prime number for n where 2^n - 1 is not prime. As we saw in the exercise, the prime number 11 was used to calculate 2^{11} - 1, which equals 2047. Upon factoring, we find that 2047 is divisible by 23 and 89, showing that it is not a prime number. Thus, this single counterexample is enough to refute the original claim.

Counterexample is a powerful tool in mathematics because it requires only one instance to disprove a general statement, whereas proving a statement generally requires showing that it holds true in all cases. The ability to find counterexamples is an important skill in mathematical problem-solving and theory development.