Question

James says that if a prime number is doubled, then that new number is still a prime number. State whether you agree or disagree, and explain why you think so (e.g. you may refer to a definition). If you agree, give an example that proves James's conjecture. If you disagree, give a counterexample that disproves James's conjecture.

Answer 1

[AI-generated answer — Bot GPT 3.5]

I disagree with James's statement. While there are prime numbers that, when doubled, result in another prime number, it is not always the case. By definition, a prime number is a natural number greater than one that has no positive divisors other than 1 and itself.

A counterexample to James's conjecture is the prime number 3. If we double 3, we get 6, which is not a prime number since it has divisors other than 1 and 6 (specifically, 2 and 3). Therefore, doubling a prime number does not necessarily result in another prime number.

However, it is worth mentioning that there are prime numbers that, when doubled, do yield another prime number. For example, if we take the prime number 5 and double it, we get 10, which is not prime. Yet, if we take the prime number 7 and double it, we get 14, which is not prime either. Therefore, while there are instances where doubling a prime number results in another prime number, it is not a universally applicable statement.