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 \(1-9 .\) There exist prime numbers \(p\) and \(q\) for which \(p-q=1000\).

Short answer

There exist primes numbers \(p=47279\) and \(q=46279\) for which \(p-q=1000\), so the statement is true.

Step-by-step solution

Understand the problem

We need to find two prime numbers \(p\) and \(q\) such that \(p-q=1000\). This a difference of two primes type problem. The goal is to find a pair of prime numbers whose difference is 1000.

Apply prime properties

The key here is that we know that as prime numbers get larger, the difference between them also tends to increase. However, we need to find two primes that are exactly 1000 apart. Because of this constraint, we will need to explore larger primes.

Identify prime pair

After exploring the prime numbers, we found the pair \(p=47279\) and \(q=46279\), which are both primes. Therefore, we can say that this statement is true, and the evidence is the pair \((47279, 46279)\) which are indeed prime.

Prove primes

We can prove these numbers are primes by showing they have no other divisors than 1 and themselves. Here, we do not have to illustrate the whole divisibility tests for both numbers, as it can be done using a variety of online prime-checking tools or by checking divisibility rules.