Problem 35 - Book of Proof

Question

Each of the following statements is either true or false. If a statement is true, prove it. If a statement is false, disprove it. These exercises are cumulative, covering all topics addressed in Chapters \(1-9 .\) In Chapter 5, Exercise 25 asked you to prove that if \(2^{n}-1\) is prime, then \(n\) is prime. Is the converse true?

Step-by-Step Solution

Answer

No, the converse is not true. There exist prime \(n\) for which \(2^{n}-1\) is not a prime.

Step 1: Understand original and converse statements

The original statement says that 'if \(2^{n}-1\) is prime, then \(n\) is prime'. The converse of this statement is 'if \(n\) is prime, then \(2^{n}-1\) is prime'.

Step 2: Test the converse

To test if the converse holds true, let's analyse it for a few prime \(n\). For example, when \(n=2\) and \(n=3\), \(2^{2}-1 = 3\) and \(2^{3}-1=7\) are both prime numbers. So, as of now, the converse seems true.

Step 3: Find counterexample

To prove it wrong, one counterexample is enough. Consider when \(n = 11\) (which is prime). Now, \(2^{11} - 1 = 2047\), and 2047 = 23 * 89, which is not a prime number. Hence, even though \(n = 11\) is prime, \(2^{11}-1\) is not. Thus, the converse does not hold true for all prime \(n\).

Key Concepts

Counterexample

Understanding the concept of a counterexample is paramount in mathematical proofs. A 'counterexample' is a specific case for which a general statement is false. It works effectively to disprove conditional statements, because according to the logic within mathematics, a statement is considered false if just one instance can be found where it doesn't hold up. This approach is particularly useful because, instead of proving that a statement is false for all cases (which can be exhaustive or even impossible), finding a single counterexample is sufficient.

Let's consider the exercise where the original statement was that if a number of the form \(2^{n}-1\) is prime, then \(n\) must also be prime. The converse would assert that if \(n\) is prime, then \(2^{n}-1\) must be prime as well. To evaluate this, we didn't need to check every prime number in existence; instead, we only had to find a single prime number \(n\) where \(2^{n}-1\) wasn't prime. In this instance, that number was \(11\), proving the converse false by counterexample.

Therefore, in situations where students face difficult universal generalizations, the strategic use of a counterexample can simplify the process significantly. It's a powerful tool which allows mathematicians to affirm the limit of an assertion's truth.

Prime Numbers

Prime numbers are the building blocks of our number system, often referred to as the 'atoms' of mathematics. A prime number is a positive integer greater than 1 that has no positive integer divisors other than 1 and itself. This means that a prime number can only be divided evenly by 1 and the number itself without leaving a remainder.

The concept of prime numbers is fundamental in various areas of mathematics, including number theory and cryptography. Primes are particularly fascinating because they appear to be distributed randomly among the integers, yet they also follow certain patterns which mathematicians are still striving to fully understand.

Prime numbers are also at the heart of the exercise in question. The property that \(2^{n}-1\) is prime if \(n\) is prime closely relates to Mersenne primes, which are prime numbers of that form. However, not every prime \(n\) leads to a Mersenne prime, as the counterexample with \(n=11\) shows. It's crucial for students to recognize that prime numbers cannot be assumed to behave similarly in all situations or formulas, which also brings to light the importance of careful scrutiny in mathematical statements and proofs.

Converse Statements

A 'converse statement' is formed by reversing the direction of a conditional statement. This means if an original conditional statement was 'if P, then Q,' the converse would be 'if Q, then P.' Interestingly, the truth of the original statement does not imply the truth of its converse. In formal logic, these are considered as separate propositions and must be verified independently.

In mathematics, proving the converse of a statement requires separate verification and sometimes it is mistakenly assumed that the converse of a true statement must also be true. However, as seen in the provided exercise, the converse may not always hold true. This misstep is a common error for students first experiencing proofs and propositional logic. They can improve their mathematical reasoning skills by acknowledging that they must test the converse statement as rigorously as the original.

The takeaway for students is that one should not assume the validity of a converse statement without investigation, as this can lead to erroneous conclusions. Developing the ability to critically determine the truth of converse statements is an essential skill in mathematics and can lead to deeper insights into the nature of logical implications.